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class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/symbol.html" aria-label="常用数学符号"><!--[--><!--[--><!--]--><!--]-->常用数学符号<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/group.html" aria-label="群"><!--[--><!--[--><!--]--><!--]-->群<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_1.html" aria-label="数论（一）"><!--[--><!--[--><!--]--><!--]-->数论（一）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link route-link-active auto-link vp-sidebar-item active" href="/note/math/number_theory_2.html" aria-label="数论（二）"><!--[--><!--[--><!--]--><!--]-->数论（二）<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_5-二次剩余" 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href="#_5-5-举例" aria-label="5.5 举例"><!--[--><!--[--><!--]--><!--]-->5.5 举例<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_5-6-定理" aria-label="5.6 定理"><!--[--><!--[--><!--]--><!--]-->5.6 定理<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_5-7-举例" aria-label="5.7 举例"><!--[--><!--[--><!--]--><!--]-->5.7 举例<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_5-8-定理" aria-label="5.8 定理"><!--[--><!--[--><!--]--><!--]-->5.8 定理<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_5-9-举例" aria-label="5.9 举例"><!--[--><!--[--><!--]--><!--]-->5.9 举例<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="#_6-勒让德-雅可比符号" aria-label="6. 勒让德-雅可比符号"><!--[--><!--[--><!--]--><!--]-->6. 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class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_7-1-定理" aria-label="7.1 定理"><!--[--><!--[--><!--]--><!--]-->7.1 定理<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_7-2-举例" aria-label="7.2 举例"><!--[--><!--[--><!--]--><!--]-->7.2 举例<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_7-3-算法-求模为质数时的平方根" aria-label="7.3 算法：求模为质数时的平方根"><!--[--><!--[--><!--]--><!--]-->7.3 算法：求模为质数时的平方根<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="#_8-模合数的二次平方根" aria-label="8. 模合数的二次平方根"><!--[--><!--[--><!--]--><!--]-->8. 模合数的二次平方根<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_8-1-算法-模2的幂" aria-label="8.1 算法 模2的幂"><!--[--><!--[--><!--]--><!--]-->8.1 算法 模2的幂<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_8-2-算法-模质数的幂" aria-label="8.2 算法 模质数的幂"><!--[--><!--[--><!--]--><!--]-->8.2 算法 模质数的幂<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_8-3-举例" aria-label="8.3 举例"><!--[--><!--[--><!--]--><!--]-->8.3 举例<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_8-4-算法" aria-label="8.4 算法"><!--[--><!--[--><!--]--><!--]-->8.4 算法<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_8-5-举例" aria-label="8.5 举例"><!--[--><!--[--><!--]--><!--]-->8.5 举例<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_8-6-定理" aria-label="8.6 定理"><!--[--><!--[--><!--]--><!--]-->8.6 定理<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_3.html" aria-label="数论（三）"><!--[--><!--[--><!--]--><!--]-->数论（三）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/probability.html" aria-label="概率"><!--[--><!--[--><!--]--><!--]-->概率<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">密码学 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/rsa.html" aria-label="RSA"><!--[--><!--[--><!--]--><!--]-->RSA<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/flip_coin.html" aria-label="抛币协议"><!--[--><!--[--><!--]--><!--]-->抛币协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/mental_poker.html" aria-label="智能扑克协议"><!--[--><!--[--><!--]--><!--]-->智能扑克协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">图形学 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">数学基础 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_01.html" aria-label="矢量"><!--[--><!--[--><!--]--><!--]-->矢量<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_02.html" aria-label="矩阵"><!--[--><!--[--><!--]--><!--]-->矩阵<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_03.html" aria-label="立体角"><!--[--><!--[--><!--]--><!--]-->立体角<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_01.html" aria-label="几何变换(一)"><!--[--><!--[--><!--]--><!--]-->几何变换(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_02.html" aria-label="几何变换(二)"><!--[--><!--[--><!--]--><!--]-->几何变换(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_03.html" aria-label="法线变换"><!--[--><!--[--><!--]--><!--]-->法线变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_04.html" aria-label="摄像机变换"><!--[--><!--[--><!--]--><!--]-->摄像机变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">光照模型 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/classic/illumination_model_01.html" aria-label="传统光照模型"><!--[--><!--[--><!--]--><!--]-->传统光照模型<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/luminosity/luminosity.html" aria-label="光度学"><!--[--><!--[--><!--]--><!--]-->光度学<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/BRDF/BRDF.html" aria-label="双向反射分布函数(BRDF)"><!--[--><!--[--><!--]--><!--]-->双向反射分布函数(BRDF)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_1.html" aria-label="微平面理论(一)"><!--[--><!--[--><!--]--><!--]-->微平面理论(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_2.html" aria-label="微平面理论(二)"><!--[--><!--[--><!--]--><!--]-->微平面理论(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_3.html" aria-label="微平面理论(三)"><!--[--><!--[--><!--]--><!--]-->微平面理论(三)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/render_function/render_function_1.html" aria-label="光照方程"><!--[--><!--[--><!--]--><!--]-->光照方程<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">环境光渲染 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_01.html" aria-label="环境光渲染(一)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_02.html" aria-label="环境光渲染(二)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">编程语言 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">JavaScript <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/helloworld.html" aria-label="环境搭建"><!--[--><!--[--><!--]--><!--]-->环境搭建<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/grammar.html" aria-label="基本语法"><!--[--><!--[--><!--]--><!--]-->基本语法<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/function.html" aria-label="函数"><!--[--><!--[--><!--]--><!--]-->函数<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/object_and_class.html" aria-label="对象和类"><!--[--><!--[--><!--]--><!--]-->对象和类<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul><!--[--><!--]--></aside><!--]--><!--[--><main class="vp-page"><!--[--><!--]--><div vp-content><!--[--><!--]--><div><h1 id="数论-二" tabindex="-1"><a class="header-anchor" href="#数论-二"><span>数论（二）</span></a></h1><hr><h2 id="_5-二次剩余" tabindex="-1"><a class="header-anchor" href="#_5-二次剩余"><span>5.二次剩余</span></a></h2><h3 id="_5-1-二次剩余-quadratic-residue-和二次非剩余定义" tabindex="-1"><a class="header-anchor" href="#_5-1-二次剩余-quadratic-residue-和二次非剩余定义"><span>5.1 二次剩余(<a href="https://en.wikipedia.org/wiki/Quadratic_residue" target="_blank" rel="noopener noreferrer">Quadratic residue</a>)和二次非剩余定义</span></a></h3><p>设整数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3E"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>></mo><mn>1</mn><mo>,</mo><mi>a</mi><mo><</mo><mi>n</mi></math></mjx-assistive-mml></mjx-container>,如果存在<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-utext variant="double-struck">𝕫</mjx-utext></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">z</mi></mrow><mi>n</mi></msub></math></mjx-assistive-mml></mjx-container>,满足<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>≡</mo><mi>a</mi><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mi>n</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>,那么称<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></mjx-assistive-mml></mjx-container>是模<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container>的二次剩余,否则<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></mjx-assistive-mml></mjx-container>就称为模<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container>的二次非剩余，用<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><msub><mi>R</mi><mi>n</mi></msub></math></mjx-assistive-mml></mjx-container>表示模<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container>的二次剩余集合，用<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mi>N</mi><msub><mi>R</mi><mi>n</mi></msub></math></mjx-assistive-mml></mjx-container>表示模<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container>的二次非剩余集合</p><h3 id="_5-2-举例" tabindex="-1"><a class="header-anchor" href="#_5-2-举例"><span>5.2 举例</span></a></h3><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:37.032em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-msup><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><mjx-mn class="mjx-n"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><mjx-mn class="mjx-n"><mjx-c class="mjx-c36"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><mjx-mn class="mjx-n"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><mjx-mn class="mjx-n"><mjx-c class="mjx-c38"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><mjx-mn class="mjx-n"><mjx-c class="mjx-c39"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2026"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo><mjx-mspace style="width:1em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c39"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c36"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c38"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mi>Q</mi><msub><mi>R</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub></mtd><mtd><mi></mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><msup><mn>0</mn><mn>2</mn></msup><mo>,</mo><msup><mn>1</mn><mn>2</mn></msup><mo>,</mo><msup><mn>2</mn><mn>2</mn></msup><mo>,</mo><msup><mn>3</mn><mn>2</mn></msup><mo>,</mo><msup><mn>4</mn><mn>2</mn></msup><mo>,</mo><msup><mn>5</mn><mn>2</mn></msup><mo>,</mo><msup><mn>6</mn><mn>2</mn></msup><mo>,</mo><msup><mn>7</mn><mn>2</mn></msup><mo>,</mo><msup><mn>8</mn><mn>2</mn></msup><mo>,</mo><msup><mn>9</mn><mn>2</mn></msup><mo>,</mo><msup><mn>10</mn><mn>2</mn></msup><mo>,</mo><msup><mn>11</mn><mn>2</mn></msup><mo>,</mo><mo>…</mo><mo fence="false" stretchy="false">}</mo><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>11</mn><mo stretchy="false">)</mo><mo>=</mo><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>9</mn><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>Q</mi><mi>N</mi><msub><mi>R</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub></mtd><mtd><mi></mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>10</mn><mo fence="false" stretchy="false">}</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><h3 id="_5-3-定理-欧拉准则" tabindex="-1"><a class="header-anchor" href="#_5-3-定理-欧拉准则"><span>5.3 定理：欧拉准则</span></a></h3><p>针对质数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></mjx-assistive-mml></mjx-container>的二次剩余判定，对于质数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></mjx-assistive-mml></mjx-container>,任意<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-msubsup space="4"><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-spacer style="margin-top:0.322em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><msubsup><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow><mi>p</mi><mrow data-mjx-texclass="ORD"><mo>∗</mo></mrow></msubsup></math></mjx-assistive-mml></mjx-container>，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi>Q</mi><msub><mi>R</mi><mi>p</mi></msub></math></mjx-assistive-mml></mjx-container>的充分必要条件为</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mspace style="width:1em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msup><mo>≡</mo><mn>1</mn><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mi>p</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><p><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi>Q</mi><mi>N</mi><msub><mi>R</mi><mi>p</mi></msub></math></mjx-assistive-mml></mjx-container>的充分必要条件为</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mspace style="width:1em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msup><mo>≡</mo><mo>−</mo><mn>1</mn><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mi>p</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><h3 id="_5-4-定理" tabindex="-1"><a class="header-anchor" href="#_5-4-定理"><span>5.4 定理</span></a></h3><p>针对合数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container>的二次剩余判定，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-TeXAtom texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-TeXAtom><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-msup space="3"><mjx-TeXAtom texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-TeXAtom><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22EF"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-msup space="3"><mjx-TeXAtom texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-TeXAtom><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-TeXAtom></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><msup><mrow data-mjx-texclass="ORD"><msub><mi>p</mi><mn>1</mn></msub></mrow><mrow data-mjx-texclass="ORD"><msub><mi>a</mi><mn>1</mn></msub></mrow></msup><mo>×</mo><msup><mrow data-mjx-texclass="ORD"><msub><mi>p</mi><mn>1</mn></msub></mrow><mrow data-mjx-texclass="ORD"><msub><mi>a</mi><mn>1</mn></msub></mrow></msup><mo>×</mo><mo>⋯</mo><mo>×</mo><msup><mrow data-mjx-texclass="ORD"><msub><mi>p</mi><mi>k</mi></msub></mrow><mrow data-mjx-texclass="ORD"><msub><mi>a</mi><mi>k</mi></msub></mrow></msup></math></mjx-assistive-mml></mjx-container>，那么<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi>Q</mi><msub><mi>R</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub></math></mjx-assistive-mml></mjx-container>的充分必要条件为：</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mspace style="width:1em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-msup space="2"><mjx-TeXAtom texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-TeXAtom><mjx-script style="vertical-align:0.413em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-msup><mjx-TeXAtom texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-TeXAtom><mjx-script style="vertical-align:0.336em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.307em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-TeXAtom></mjx-script></mjx-msup></mjx-TeXAtom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>x</mi><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><msup><mrow data-mjx-texclass="ORD"><msub><mi>p</mi><mi>i</mi></msub></mrow><mrow data-mjx-texclass="ORD"><msub><mi>a</mi><mi>i</mi></msub></mrow></msup><mo stretchy="false">)</mo><mo>∈</mo><mi>Q</mi><msub><mi>R</mi><mrow data-mjx-texclass="ORD"><msup><mrow data-mjx-texclass="ORD"><msub><mi>p</mi><mi>i</mi></msub></mrow><mrow data-mjx-texclass="ORD"><msub><mi>a</mi><mi>i</mi></msub></mrow></msup></mrow></msub></math></mjx-assistive-mml></mjx-container><h3 id="_5-5-举例" tabindex="-1"><a class="header-anchor" href="#_5-5-举例"><span>5.5 举例</span></a></h3><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:20.099em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c37"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c37"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c36"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c39"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c36"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c39"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mi>n</mi></mtd><mtd><mi></mi><mo>=</mo><mn>5</mn><mo>×</mo><mn>7</mn></mtd></mtr><mtr><mtd><mi>Q</mi><msub><mi>R</mi><mn>5</mn></msub></mtd><mtd><mi></mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>Q</mi><mi>N</mi><msub><mi>R</mi><mn>5</mn></msub></mtd><mtd><mi></mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>Q</mi><msub><mi>R</mi><mn>7</mn></msub></mtd><mtd><mi></mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>Q</mi><mi>N</mi><msub><mi>R</mi><mn>7</mn></msub></mtd><mtd><mi></mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>Q</mi><msub><mi>R</mi><mrow data-mjx-texclass="ORD"><mn>35</mn></mrow></msub></mtd><mtd><mi></mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>9</mn><mo>,</mo><mn>11</mn><mo>,</mo><mn>14</mn><mo>,</mo><mn>15</mn><mo>,</mo><mn>16</mn><mo>,</mo><mn>21</mn><mo>,</mo><mn>25</mn><mo>,</mo><mn>29</mn><mo>,</mo><mn>30</mn><mo fence="false" stretchy="false">}</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><h3 id="_5-6-定理" tabindex="-1"><a class="header-anchor" href="#_5-6-定理"><span>5.6 定理</span></a></h3><p>对于质数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></mjx-assistive-mml></mjx-container>,<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msubsup><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-n" size="s"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.322em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow><mi>p</mi><mo>∗</mo></msubsup></math></mjx-assistive-mml></mjx-container>中有<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn></math></mjx-assistive-mml></mjx-container>个元素是模<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></mjx-assistive-mml></mjx-container>的二次剩余，另外<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn></math></mjx-assistive-mml></mjx-container>个元素是模<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></mjx-assistive-mml></mjx-container>的二次非剩余</p><h3 id="_5-7-举例" tabindex="-1"><a class="header-anchor" href="#_5-7-举例"><span>5.7 举例</span></a></h3><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msubsup><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-spacer style="margin-top:0.214em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c36"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c38"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c39"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo><mjx-mspace></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c39"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mspace></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c36"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c38"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mspace></mjx-mspace></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow><mrow data-mjx-texclass="ORD"><mo>∗</mo></mrow></msubsup><mo>=</mo><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>9</mn><mo>,</mo><mn>10</mn><mo fence="false" stretchy="false">}</mo><mspace linebreak="newline"></mspace><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>9</mn><mo fence="false" stretchy="false">}</mo><mo>∈</mo><mi>Q</mi><msub><mi>R</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub><mspace linebreak="newline"></mspace><mo fence="false" stretchy="false">{</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>10</mn><mo fence="false" stretchy="false">}</mo><mo>∈</mo><mi>Q</mi><mi>N</mi><msub><mi>R</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub><mspace linebreak="newline"></mspace></math></mjx-assistive-mml></mjx-container><h3 id="_5-8-定理" tabindex="-1"><a class="header-anchor" href="#_5-8-定理"><span>5.8 <strong>定理</strong></span></a></h3><p>对于两个质数的乘积<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>q</mi><mi>p</mi></math></mjx-assistive-mml></mjx-container>，那么<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub></math></mjx-assistive-mml></mjx-container>里恰好有1/4的数，也就是<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>4</mn></math></mjx-assistive-mml></mjx-container>个是二次剩余，一般来说，如果合数n有k个质数因子，那么<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msubsup><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-spacer style="margin-top:0.322em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msubsup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow><mrow data-mjx-texclass="ORD"><mo>∗</mo></mrow></msubsup></math></mjx-assistive-mml></mjx-container>中的元素中有<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-msup><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><msup><mn>2</mn><mi>k</mi></msup></math></mjx-assistive-mml></mjx-container>是二次剩余</p><h3 id="_5-9-举例" tabindex="-1"><a class="header-anchor" href="#_5-9-举例"><span>5.9 <strong>举例</strong></span></a></h3><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msubsup><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c2124 TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-spacer style="margin-top:0.214em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c38"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo><mjx-mspace></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mspace></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c38"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mspace></mjx-mspace></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">Z</mi></mrow><mrow data-mjx-texclass="ORD"><mn>15</mn></mrow><mrow data-mjx-texclass="ORD"><mo>∗</mo></mrow></msubsup><mo>=</mo><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>11</mn><mo>,</mo><mn>13</mn><mo>,</mo><mn>14</mn><mo fence="false" stretchy="false">}</mo><mspace linebreak="newline"></mspace><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo fence="false" stretchy="false">}</mo><mo>∈</mo><mi>Q</mi><msub><mi>R</mi><mrow data-mjx-texclass="ORD"><mn>15</mn></mrow></msub><mspace linebreak="newline"></mspace><mo fence="false" stretchy="false">{</mo><mn>2</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>11</mn><mo>,</mo><mn>13</mn><mo>,</mo><mn>14</mn><mo fence="false" stretchy="false">}</mo><mo>∈</mo><mi>Q</mi><mi>N</mi><msub><mi>R</mi><mrow data-mjx-texclass="ORD"><mn>15</mn></mrow></msub><mspace linebreak="newline"></mspace></math></mjx-assistive-mml></mjx-container><h2 id="_6-勒让德-雅可比符号" tabindex="-1"><a class="header-anchor" href="#_6-勒让德-雅可比符号"><span>6. 勒让德-雅可比符号</span></a></h2><h3 id="_6-1-定义" tabindex="-1"><a class="header-anchor" href="#_6-1-定义"><span>6.1 定义</span></a></h3><p>对于任意质数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3E"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>></mo><mn>2</mn></math></mjx-assistive-mml></mjx-container>，和任意<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></mjx-assistive-mml></mjx-container>，定义勒让德符号(<a href="https://en.wikipedia.org/wiki/Legendre_symbol" target="_blank" rel="noopener noreferrer">Legendre_symbol</a>)为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.474em;vertical-align:-1.487em;" class="mjx-c7B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-mid><mjx-c></mjx-c></mjx-mid><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:16.328em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-bottom:0.1em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-bottom:0.1em;"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c66"></mjx-c><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2262"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-top:0.1em;padding-bottom:0.1em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-top:0.1em;padding-bottom:0.1em;"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c66"></mjx-c><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2262"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-top:0.1em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-top:0.1em;"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c66"></mjx-c><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n" style="vertical-align:0.25em;"></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>L</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mtable columnalign="left left" columnspacing="1em" rowspacing=".2em"><mtr><mtd><mn>1</mn></mtd><mtd><mtext>if </mtext><mi>a</mi><mo>∈</mo><mi>Q</mi><msub><mi>R</mi><mi>p</mi></msub><mtext> and </mtext><mi>a</mi><mo>≢</mo><mn>0</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mi>p</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>−</mo><mn>1</mn></mtd><mtd><mtext>if </mtext><mi>a</mi><mo>∈</mo><mi>Q</mi><mi>N</mi><msub><mi>R</mi><mi>p</mi></msub><mtext> and </mtext><mi>a</mi><mo>≢</mo><mn>0</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mi>p</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mtext>if </mtext><mi>a</mi><mo>≡</mo><mn>0</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mi>p</mi><mo stretchy="false">)</mo></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo></mrow></math></mjx-assistive-mml></mjx-container><h3 id="_6-2-定理" tabindex="-1"><a class="header-anchor" href="#_6-2-定理"><span>6.2 定理</span></a></h3><p>勒让德符号计算方法</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mspace style="width:1em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mtext class="mjx-n" space="2"><mjx-c class="mjx-cA0"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>L</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>a</mi><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msup><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mi>p</mi><mo stretchy="false">)</mo><mo>,</mo><mtext> and </mtext><mi>L</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=∈</mo><mo fence="false" stretchy="false">{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo fence="false" stretchy="false">}</mo></math></mjx-assistive-mml></mjx-container><h3 id="_6-3-定义" tabindex="-1"><a class="header-anchor" href="#_6-3-定义"><span>6.3 定义</span></a></h3><p>对于合数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3E"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>></mo><mn>1</mn></math></mjx-assistive-mml></mjx-container>, 其素因子<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c22EF"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><msub><mi>p</mi><mn>1</mn></msub><msub><mi>p</mi><mn>2</mn></msub><mo>⋯</mo><msub><mi>p</mi><mi>k</mi></msub></math></mjx-assistive-mml></mjx-container>（可重复），任意整数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></mjx-assistive-mml></mjx-container>，定义雅可比符号(<a href="https://en.wikipedia.org/wiki/Jacobi_symbol" target="_blank" rel="noopener noreferrer">Jacobi symbol</a>)</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c22EF"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>J</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>L</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msub><mi>p</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>L</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msub><mi>p</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>⋯</mo><mi>L</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msub><mi>p</mi><mi>k</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><p>注意：雅可比符号不能确定一个数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></mjx-assistive-mml></mjx-container>是否是对模<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container>的二次剩余（除非<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container>是质数），比如</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>J</mi><mo stretchy="false">(</mo><mn>7</mn><mo>,</mo><mn>143</mn><mo stretchy="false">)</mo><mo>=</mo><mi>L</mi><mo stretchy="false">(</mo><mn>7</mn><mo>,</mo><mn>11</mn><mo stretchy="false">)</mo><mi>L</mi><mo stretchy="false">(</mo><mn>7</mn><mo>,</mo><mn>13</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container><p>但7并不是143的二次剩余</p><h3 id="_6-4-简写" tabindex="-1"><a class="header-anchor" href="#_6-4-简写"><span>6.4 简写</span></a></h3><p>勒让德符号和雅可比符号也可统一简写为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mrow><mjx-mo class="mjx-lop"><mjx-c class="mjx-c28 TEX-S2"></mjx-c></mjx-mo><mjx-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-lop"><mjx-c class="mjx-c29 TEX-S2"></mjx-c></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><mi>a</mi><mi>p</mi></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow></math></mjx-assistive-mml></mjx-container> ,或者<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>a</mi><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mi>p</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></p><h2 id="_7-模质数的二次平方根" tabindex="-1"><a class="header-anchor" href="#_7-模质数的二次平方根"><span>7. 模质数的二次平方根</span></a></h2><h3 id="_7-1-定理" tabindex="-1"><a class="header-anchor" href="#_7-1-定理"><span>7.1 定理</span></a></h3><p>对于质数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3E"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>></mo><mn>2</mn></math></mjx-assistive-mml></mjx-container>，如果<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2260"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi>Q</mi><msub><mi>R</mi><mi>p</mi></msub><mo>,</mo><mi>a</mi><mo>≠</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container>，那么方程</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mspace style="width:1em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>x</mi><mn>2</mn></msup><mo>≡</mo><mi>a</mi><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mi>p</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><p>在有两个解，也就是说a有两个平方根，其中一个在区间<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>，另一个在区间<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn><mo>,</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>，而且，其中一个平方根也属于模<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></mjx-assistive-mml></mjx-container>的二次剩余，称为主平方根(pricipal square root)</p><h3 id="_7-2-举例" tabindex="-1"><a class="header-anchor" href="#_7-2-举例"><span>7.2 举例</span></a></h3><p>对于方程</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c39"></mjx-c></mjx-mn><mjx-mspace style="width:1em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mspace></mjx-mspace></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>x</mi><mn>2</mn></msup><mo>≡</mo><mn>9</mn><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>11</mn><mo stretchy="false">)</mo><mspace linebreak="newline"></mspace></math></mjx-assistive-mml></mjx-container><p>有两个解<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c38"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>1</mn></msub><mo>=</mo><mn>3</mn><mo>,</mo><msub><mi>x</mi><mn>2</mn></msub><mo>=</mo><mn>8</mn></math></mjx-assistive-mml></mjx-container>, 其中</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c38"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c36"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mn>3</mn><mo>∈</mo><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo stretchy="false">]</mo><mo>,</mo><mn>8</mn><mo>∈</mo><mo stretchy="false">[</mo><mn>6</mn><mo>,</mo><mn>10</mn><mo stretchy="false">]</mo><mo>,</mo><mn>3</mn><mo>∈</mo><mi>Q</mi><msub><mi>R</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub></math></mjx-assistive-mml></mjx-container><h3 id="_7-3-算法-求模为质数时的平方根" tabindex="-1"><a class="header-anchor" href="#_7-3-算法-求模为质数时的平方根"><span>7.3 算法：求模为质数时的平方根</span></a></h3><h4 id="_7-3-1-特殊情况" tabindex="-1"><a class="header-anchor" href="#_7-3-1-特殊情况"><span>7.3.1 特殊情况</span></a></h4><p>对于方程<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>≡</mo><mi>a</mi><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mi>p</mi><mo stretchy="false">)</mo><mo>,</mo><mi>a</mi><mo>∈</mo><mi>Q</mi><msub><mi>R</mi><mi>p</mi></msub></math></mjx-assistive-mml></mjx-container> 如果<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c38"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>≡</mo><mn>3</mn><mo>,</mo><mn>7</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>8</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>, <mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-cB1"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≡</mo><mo>±</mo><msup><mi>a</mi><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msup><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mi>p</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> 如果<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c38"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>≡</mo><mn>5</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>8</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>, <mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-cB1"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mn class="mjx-n"><mjx-c class="mjx-c38"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≡</mo><mo>±</mo><msup><mi>a</mi><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>8</mn></mrow></msup><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mi>p</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> 如果<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>≡</mo><mn>3</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>4</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>, <mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-cB1"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≡</mo><mo>±</mo><msup><mi>a</mi><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>4</mn></mrow></msup><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mi>p</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></p><h4 id="_7-3-2" tabindex="-1"><a class="header-anchor" href="#_7-3-2"><span>7.3.2</span></a></h4><p>对于一般情况的质数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></mjx-assistive-mml></mjx-container>,使用<a href="https://en.wikipedia.org/wiki/Tonelli%E2%80%93Shanks_algorithm" target="_blank" rel="noopener noreferrer">Tonelli–Shanks</a>算法求解</p><h2 id="_8-模合数的二次平方根" tabindex="-1"><a class="header-anchor" href="#_8-模合数的二次平方根"><span>8. 模合数的二次平方根</span></a></h2><p>参考: <a href="https://www.johndcook.com/blog/quadratic_congruences/" target="_blank" rel="noopener noreferrer">https://www.johndcook.com/blog/quadratic_congruences/</a></p><h3 id="_8-1-算法-模2的幂" tabindex="-1"><a class="header-anchor" href="#_8-1-算法-模2的幂"><span>8.1 <strong>算法</strong> 模2的幂</span></a></h3><p>对于方程<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>≡</mo><mi>a</mi><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>2</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，有解<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≡</mo><mn>1</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>2</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> 方程<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-msup space="2"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>≡</mo><mi>a</mi><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><msup><mn>2</mn><mn>2</mn></msup><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，当<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≡</mo><mn>1</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>4</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>时有2个解，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-cB1"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≡</mo><mo>±</mo><mn>1</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>4</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> 方程<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-msup space="2"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2265"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c67"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>≡</mo><mi>a</mi><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><msup><mn>2</mn><mi>n</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mi>n</mi><mo>≥</mo><mn>3</mn><mo>,</mo><mo data-mjx-texclass="OP" movablelimits="true">gcd</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container>, 当<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c38"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≡</mo><mn>1</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>8</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>时有4个解，</p><h3 id="_8-2-算法-模质数的幂" tabindex="-1"><a class="header-anchor" href="#_8-2-算法-模质数的幂"><span>8.2 <strong>算法</strong> 模质数的幂</span></a></h3><p>求解方程<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-msup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3E"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c67"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>≡</mo><mi>a</mi><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><msup><mi>p</mi><mi>k</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mi>k</mi><mo>></mo><mn>0</mn><mo>,</mo><mo data-mjx-texclass="OP" movablelimits="true">gcd</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container>, 方程有解的充分必要条件是<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mrow><mjx-mo class="mjx-lop"><mjx-c class="mjx-c28 TEX-S2"></mjx-c></mjx-mo><mjx-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-lop"><mjx-c class="mjx-c29 TEX-S2"></mjx-c></mjx-mo></mjx-mrow><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><mi>a</mi><mi>p</mi></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container>, 也就是a是模p的二次剩余 方程有两个解，使用<a href="https://en.wikipedia.org/wiki/Hensel%27s_lemma" target="_blank" rel="noopener noreferrer">Hensel's lemma</a>算法 设<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>k</mi></msub></math></mjx-assistive-mml></mjx-container>是<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-msup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>≡</mo><mi>a</mi><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><msup><mi>p</mi><mi>k</mi></msup><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>的解，有<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mi>k</mi></msub></math></mjx-assistive-mml></mjx-container>存在使得等式<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-msup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msub><mi>x</mi><mi>k</mi></msub><msub><mi>y</mi><mi>k</mi></msub><mo>≡</mo><mn>1</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><msup><mi>p</mi><mi>k</mi></msup><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>成立，那么<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.308em;margin-left:0;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-spacer style="margin-top:0.18em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>k</mi></msub><mo>−</mo><mo stretchy="false">(</mo><msubsup><mi>x</mi><mi>k</mi><mn>2</mn></msubsup><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo><msub><mi>y</mi><mi>k</mi></msub></math></mjx-assistive-mml></mjx-container>是方程<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-msup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>≡</mo><mi>a</mi><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><msup><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>的解</p><h3 id="_8-3-举例" tabindex="-1"><a class="header-anchor" href="#_8-3-举例"><span>8.3 <strong>举例</strong></span></a></h3><p>求解方程</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mspace style="width:1em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>x</mi><mn>2</mn></msup><mo>≡</mo><mn>23</mn><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>343</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><p>其中<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mn class="mjx-n"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>343</mn><mo>=</mo><msup><mn>7</mn><mn>3</mn></msup></math></mjx-assistive-mml></mjx-container>，首先解方程</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-spacer style="margin-top:0.189em;"></mjx-spacer><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mspace style="width:1em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-cB1"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>x</mi><mn>1</mn><mn>2</mn></msubsup><mo>≡</mo><mn>23</mn><mo>≡</mo><mn>2</mn><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>7</mn><mo stretchy="false">)</mo><mo>,</mo><msub><mi>x</mi><mn>1</mn></msub><mo>=</mo><mo>±</mo><mn>3</mn></math></mjx-assistive-mml></mjx-container><p>以<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>1</mn></msub><mo>=</mo><mn>3</mn></math></mjx-assistive-mml></mjx-container>为例，使用扩展欧几里得算法求解</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mspace style="width:1em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c36"></mjx-c></mjx-mn><mjx-mspace></mjx-mspace><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-spacer style="margin-top:0.189em;"></mjx-spacer><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mspace style="width:1em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c39"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c38"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mn>2</mn><mo>×</mo><mn>3</mn><mo>×</mo><msub><mi>y</mi><mn>1</mn></msub><mo>≡</mo><mn>1</mn><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>7</mn><mo stretchy="false">)</mo><mo>,</mo><msub><mi>y</mi><mn>1</mn></msub><mo>=</mo><mn>6</mn><mspace linebreak="newline"></mspace><msub><mi>x</mi><mn>2</mn></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>−</mo><mo stretchy="false">(</mo><msubsup><mi>x</mi><mn>1</mn><mn>2</mn></msubsup><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo><msub><mi>y</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>49</mn><mo stretchy="false">)</mo><mo>=</mo><mn>38</mn></math></mjx-assistive-mml></mjx-container><p>同理，求方程</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c38"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mspace style="width:1em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c39"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mspace></mjx-mspace><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c38"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c38"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mspace style="width:1em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c38"></mjx-c><mjx-c class="mjx-c37"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mn>2</mn><mo>×</mo><mn>38</mn><mo>×</mo><msub><mi>y</mi><mn>2</mn></msub><mo>≡</mo><mn>1</mn><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>49</mn><mo stretchy="false">)</mo><mo>,</mo><msub><mi>y</mi><mn>2</mn></msub><mo>=</mo><mn>20</mn><mspace linebreak="newline"></mspace><msub><mi>x</mi><mn>3</mn></msub><mo>=</mo><mo stretchy="false">(</mo><mn>38</mn><mo>−</mo><mo stretchy="false">(</mo><msup><mn>38</mn><mn>2</mn></msup><mo>−</mo><mn>23</mn><mo stretchy="false">)</mo><mo>×</mo><mn>20</mn><mo stretchy="false">)</mo><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>343</mn><mo stretchy="false">)</mo><mo>=</mo><mn>87</mn></math></mjx-assistive-mml></mjx-container><p>所以，最终方程的两个解是<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-cB1"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c38"></mjx-c><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c38"></mjx-c><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c35"></mjx-c><mjx-c class="mjx-c36"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>±</mo><mn>87</mn><mo stretchy="false">(</mo><mn>87</mn><mo>,</mo><mn>256</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></p><h3 id="_8-4-算法" tabindex="-1"><a class="header-anchor" href="#_8-4-算法"><span>8.4 <strong>算法</strong></span></a></h3><p>对于合数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c22EF"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><msub><mi>p</mi><mn>1</mn></msub><msub><mi>p</mi><mn>2</mn></msub><mo>⋯</mo><msub><mi>p</mi><mi>k</mi></msub></math></mjx-assistive-mml></mjx-container>， 整数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D444 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi>Q</mi><msub><mi>R</mi><mi>n</mi></msub></math></mjx-assistive-mml></mjx-container>, 求解方程<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>≡</mo><mi>a</mi><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mi>n</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> 首先求解模质数方程</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-spacer style="margin-top:0.347em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-TeXAtom></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mspace style="width:1em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mspace></mjx-mspace><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-spacer style="margin-top:0.347em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-TeXAtom></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mspace style="width:1em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mspace></mjx-mspace><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c22EF"></mjx-c></mjx-mo><mjx-mspace></mjx-mspace><mjx-msubsup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-spacer style="margin-top:0.347em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-TeXAtom></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mspace style="width:1em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>x</mi><mrow data-mjx-texclass="ORD"><msub><mi>p</mi><mn>1</mn></msub></mrow><mn>2</mn></msubsup><mo>≡</mo><mi>a</mi><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><msub><mi>p</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mspace linebreak="newline"></mspace><msubsup><mi>x</mi><mrow data-mjx-texclass="ORD"><msub><mi>p</mi><mn>1</mn></msub></mrow><mn>2</mn></msubsup><mo>≡</mo><mi>a</mi><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><msub><mi>p</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mspace linebreak="newline"></mspace><mo>⋯</mo><mspace linebreak="newline"></mspace><msubsup><mi>x</mi><mrow data-mjx-texclass="ORD"><msub><mi>p</mi><mi>k</mi></msub></mrow><mn>2</mn></msubsup><mo>≡</mo><mi>a</mi><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><msub><mi>p</mi><mi>k</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><p>然后根据中国剩余定理，解方程组</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mrow><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:4.412em;vertical-align:-1.956em;" class="mjx-c7B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-mid><mjx-c></mjx-c></mjx-mid><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:7.8em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:left;padding-bottom:0.05em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:left;padding-top:0.05em;padding-bottom:0.05em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:left;padding-top:0.05em;padding-bottom:0.05em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c22EF"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:left;padding-top:0.05em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n" style="vertical-align:0.25em;"></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mtable rowspacing=".1em" columnspacing="1em" columnalign="left left"><mtr><mtd><mi>x</mi><mo>≡</mo><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><msub><mi>p</mi><mn>1</mn></msub></mrow></msub><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><msub><mi>p</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>x</mi><mo>≡</mo><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><msub><mi>p</mi><mn>2</mn></msub></mrow></msub><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><msub><mi>p</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>⋯</mo></mtd></mtr><mtr><mtd><mi>x</mi><mo>≡</mo><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><msub><mi>p</mi><mi>k</mi></msub></mrow></msub><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><msub><mi>p</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo></mrow></math></mjx-assistive-mml></mjx-container><p>得到<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></mjx-assistive-mml></mjx-container>，由于每个<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-TeXAtom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><msub><mi>p</mi><mi>i</mi></msub></mrow></msub></math></mjx-assistive-mml></mjx-container>有两个解，所以最终<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></mjx-assistive-mml></mjx-container>有<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mi>k</mi></msup></math></mjx-assistive-mml></mjx-container>个解</p><h3 id="_8-5-举例" tabindex="-1"><a class="header-anchor" href="#_8-5-举例"><span>8.5 <strong>举例</strong></span></a></h3><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mspace style="width:1em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c37"></mjx-c><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>x</mi><mn>2</mn></msup><mo>≡</mo><mn>15</mn><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>77</mn><mo stretchy="false">)</mo><mo>,</mo><mi>p</mi><mo>=</mo><mn>7</mn><mo>,</mo><mi>q</mi><mo>=</mo><mn>11</mn></math></mjx-assistive-mml></mjx-container><p>按照模为质数的方法解方程</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mrow><mjx-mo class="mjx-s3"><mjx-c class="mjx-c7B TEX-S3"></mjx-c></mjx-mo><mjx-mtable style="min-width:16.881em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-bottom:0.1em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-spacer style="margin-top:0.297em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-bottom:0.1em;"><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-cB1"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c36"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-top:0.1em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-spacer style="margin-top:0.297em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-top:0.1em;"><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-cB1"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c39"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c39"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n" style="vertical-align:0.25em;"></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mtable columnalign="left left" columnspacing="1em" rowspacing=".2em"><mtr><mtd><msubsup><mi>x</mi><mi>p</mi><mn>2</mn></msubsup><mo>≡</mo><mn>15</mn><mo>≡</mo><mn>1</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>7</mn><mo stretchy="false">)</mo></mtd><mtd><mtext> </mtext><msub><mi>x</mi><mi>p</mi></msub><mo>=</mo><mo>±</mo><mn>1</mn><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mi>x</mi><mi>q</mi><mn>2</mn></msubsup><mo>≡</mo><mn>15</mn><mo>≡</mo><mn>4</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>11</mn><mo stretchy="false">)</mo></mtd><mtd><mtext> </mtext><msub><mi>x</mi><mi>q</mi></msub><mo>=</mo><mo>±</mo><mn>9</mn><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>9</mn><mo stretchy="false">)</mo></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo></mrow></math></mjx-assistive-mml></mjx-container><p>按照中国剩余定理，解以下四个方程组</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mrow><mjx-mo class="mjx-s3"><mjx-c class="mjx-c7B TEX-S3"></mjx-c></mjx-mo><mjx-mtable style="min-width:7.016em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:left;padding-bottom:0.05em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:left;padding-top:0.05em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n" style="vertical-align:0.25em;"></mjx-mo></mjx-mrow><mjx-mrow space="2"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c7B TEX-S3"></mjx-c></mjx-mo><mjx-mtable style="min-width:7.016em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:left;padding-bottom:0.05em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:left;padding-top:0.05em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c39"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n" style="vertical-align:0.25em;"></mjx-mo></mjx-mrow><mjx-mrow space="2"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c7B TEX-S3"></mjx-c></mjx-mo><mjx-mtable style="min-width:7.016em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:left;padding-bottom:0.05em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c36"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:left;padding-top:0.05em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n" style="vertical-align:0.25em;"></mjx-mo></mjx-mrow><mjx-mrow space="2"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c7B TEX-S3"></mjx-c></mjx-mo><mjx-mtable style="min-width:7.016em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:left;padding-bottom:0.05em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c36"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:left;padding-top:0.05em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c39"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n" style="vertical-align:0.25em;"></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mtable rowspacing=".1em" columnspacing="1em" columnalign="left left"><mtr><mtd><mi>x</mi><mo>≡</mo><mn>1</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>7</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>x</mi><mo>≡</mo><mn>2</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>11</mn><mo stretchy="false">)</mo></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mtable rowspacing=".1em" columnspacing="1em" columnalign="left left"><mtr><mtd><mi>x</mi><mo>≡</mo><mn>1</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>7</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>x</mi><mo>≡</mo><mn>9</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>11</mn><mo stretchy="false">)</mo></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mtable rowspacing=".1em" columnspacing="1em" columnalign="left left"><mtr><mtd><mi>x</mi><mo>≡</mo><mn>6</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>7</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>x</mi><mo>≡</mo><mn>2</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>11</mn><mo stretchy="false">)</mo></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mtable rowspacing=".1em" columnspacing="1em" columnalign="left left"><mtr><mtd><mi>x</mi><mo>≡</mo><mn>6</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>7</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>x</mi><mo>≡</mo><mn>9</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>11</mn><mo stretchy="false">)</mo></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo></mrow></math></mjx-assistive-mml></mjx-container><p>得到四个解<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c35"></mjx-c><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c36"></mjx-c><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>57</mn><mo>,</mo><mn>64</mn><mo>,</mo><mn>13</mn><mo>,</mo><mn>20</mn></math></mjx-assistive-mml></mjx-container></p><h3 id="_8-6-定理" tabindex="-1"><a class="header-anchor" href="#_8-6-定理"><span>8.6 <strong>定理</strong></span></a></h3><p>如果不知道<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container>的素因子，那么就模<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container>的平方根是非常困难的问题</p></div><!--[--><!--]--></div><footer class="vp-page-meta"><!----><div class="vp-meta-item git-info"><div class="vp-meta-item last-updated"><span class="meta-item-label">Last Updated: </span><!----></div><div class="vp-meta-item contributors"><span class="meta-item-label">Contributors: </span><span class="meta-item-info"><!--[--><!--[--><span class="contributor" title="email: thejinchao@gmail.com">thejinchao</span><!----><!--]--><!--]--></span></div></div></footer><nav class="vp-page-nav" aria-label="page navigation"><a class="route-link auto-link prev" href="/note/math/number_theory_1.html" aria-label="数论（一）"><!--[--><div class="hint"><span class="arrow left"></span> Prev</div><div class="link"><span>数论（一）</span></div><!--]--></a><a 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