「学习总结」数论再探

📅 Last Updated: 2021-03-06
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结束了「学习总结」整数模 n 乘法群 之后，发现数论还有很多东西妹学。 (T__T). 一直没有发表

数论初探

「学习总结」数论

整数模 n 乘法群

「学习总结」整数模 n 乘法群

库默尔定理

令 Vp(n)V_p(n) 表示 nn 中素因子 PP 的次数，有： Vp((nm)) V_p\left(\dbinom{n}{m}\right) 的值为 mm 与 n−mn-m 在 PP 进制下加法的进位次数。

另一个事实是，n!n! 中质因子 p​p​ 的幂次为： n−fp(n)p−1 \frac{n-f_p(n)}{p-1} fp(n)f_p(n) 为 nn 在 pp 进制下的数位和。

二次剩余

一个数 aa，如果不是 pp 的倍数且模 pp 同余于某个数的平方，则称 aa 为模 pp 的 二次剩余。

而一个不是 pp 的倍数的数 bb，不同余于任何数的平方，则称 bb 为模 pp 的 非二次剩余。

勒让德符号

(np)={1p∤n且n是p的二次剩余−1p∤n且n不是p的二次剩余0p∣n \left(\frac{n}{p}\right)= \begin{cases} 1 &p\nmid n \text{且}n\text{是}p\text{的二次剩余}\\ -1&p\nmid n \text{且}n\text{不是}p\text{的二次剩余}\\ 0&p\mid n \end{cases}

欧拉准则

(np)≡np−12(modp) \left(\frac{n}{p}\right)\equiv n^{\frac{p-1}{2}}\pmod p

剩余系开根

Cipolla算法

给出正整数 nn 求出模 pp 意义下的 n\sqrt n。

背过程：

随机一个 nn 的剩余系下整数 aa ，满足 a2−na^2-n 不是二次剩余。

设剩余系下虚数单位 i=a2−n\mathbf{i}=\sqrt{a^2-n}

则 n≡(a+i)p+12(modp)\sqrt n \equiv (a + \mathbf{i})^{\frac{p+1}{2}} \pmod{p}

一道例题

Code+#7 同余方程

int np[_], prime[_], tot, d[_], x;
void Prime(int n) {
    for(int i = 2; i <= n; i++){
        if(!np[i]) prime[++tot] = i, d[i] = i;
        for(int j = 1; j <= tot && (x = prime[j] * i) <= n; j++){
            np[x] = 1; d[x] = prime[j];
            if(i % prime[j] == 0) break;
        }
    }
}
pair<int, int> IN[_];
int main(){
    int n; IO >> n; int MAX = 0;
    for(int i = 1; i <= n; i++) (IO >> IN[i].first >> IN[i].second), MAX = max(MAX, IN[i].first) ;
    Prime(MAX + 1);
    for(int i = 1; i <= n; i++) {
        int p = IN[i].first, x = IN[i].second; LL ret = 1;
        while(p != 1){
            int now = d[p];
            if(x % now == 0) ret *= (now % 4 == 1 ? ((now << 1) - 1) : 1);
            else ret *= (((now % 4 == 1) ? -1 : 1) + now);
            p /= d[p];
        }
        printf("%lld\n", ret);
    }
    return 0;
}

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🔗 Source Hash: b29df85416842943fa38b88dc4285636b8f6aa31939b64913e1fd42045236e5c
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    pdftitle={「学习总结」数论再探},
    pdfauthor={Jiayi Su (ShuYuMo)},
    hidelinks,
    pdfcreator={LaTeX via pandoc}}
  
  \title{「学习总结」数论再探}
  \author{Jiayi Su (ShuYuMo)}
  \date{2021-03-06 18:58:03}
  
  \begin{document}
  \maketitle
  
  \setstretch{1.3}
  结束了\href{/「学习总结」整数模\%20n\%20乘法群}{「学习总结」整数模 n
  乘法群} 之后，发现数论还有很多东西妹学。 (T\_\_T). 一直没有发表
  
  \subsection{数论初探}\label{ux6570ux8bbaux521dux63a2}
  
  \href{/「学习总结」数论}{「学习总结」数论}
  
  \subsection{整数模 n
  乘法群}\label{ux6574ux6570ux6a21-n-ux4e58ux6cd5ux7fa4}
  
  \href{/「学习总结」整数模\%20n\%20乘法群}{「学习总结」整数模 n 乘法群}
  
  \subsection{库默尔定理}\label{ux5e93ux9ed8ux5c14ux5b9aux7406}
  
  令 \(V_p(n)\) 表示 \(n\) 中素因子 \(P\) 的次数，有： \[
  V_p\left(\dbinom{n}{m}\right)
  \] 的值为 \(m\) 与 \(n-m\) 在 \(P\) 进制下加法的进位次数。
  
  另一个事实是，\(n!\) 中质因子 \(p​\) 的幂次为： \[
  \frac{n-f_p(n)}{p-1}
  \] \(f_p(n)\) 为 \(n\) 在 \(p\) 进制下的数位和。
  
  \subsection{二次剩余}\label{ux4e8cux6b21ux5269ux4f59}
  
  一个数 \(a\)，如果不是 \(p\) 的倍数且模 \(p\) 同余于某个数的平方，则称
  \(a\) 为模 \(p\) 的 \textbf{二次剩余}。
  
  而一个不是 \(p\) 的倍数的数 \(b\)，不同余于任何数的平方，则称 \(b\) 为模
  \(p\) 的 \textbf{非二次剩余}。
  
  \subsubsection{勒让德符号}\label{ux52d2ux8ba9ux5fb7ux7b26ux53f7}
  
  \[
  \left(\frac{n}{p}\right)=
  \begin{cases} 1 &p\nmid n \text{且}n\text{是}p\text{的二次剩余}\\
  -1&p\nmid n \text{且}n\text{不是}p\text{的二次剩余}\\
  0&p\mid n 
  \end{cases}
  \]
  
  \subsubsection{欧拉准则}\label{ux6b27ux62c9ux51c6ux5219}
  
  \[
  \left(\frac{n}{p}\right)\equiv n^{\frac{p-1}{2}}\pmod p
  \]
  
  \subsubsection{剩余系开根}\label{ux5269ux4f59ux7cfbux5f00ux6839}
  
  \textbf{Cipolla算法}
  
  给出正整数 \(n\) 求出模 \(p\) 意义下的 \(\sqrt n\)。
  
  背过程：
  
  随机一个 \(n\) 的剩余系下整数 \(a\) ，满足 \(a^2-n\) 不是二次剩余。
  
  设剩余系下虚数单位 \(\mathbf{i}=\sqrt{a^2-n}\)
  
  则 \(\sqrt n \equiv (a + \mathbf{i})^{\frac{p+1}{2}} \pmod{p}\)
  
  \paragraph{一道例题}\label{ux4e00ux9053ux4f8bux9898}
  
  \href{https://www.luogu.com.cn/problem/P6610}{Code+\#7 同余方程}
  
  \begin{Shaded}
  \begin{Highlighting}[]
  \DataTypeTok{int}\NormalTok{ np}\OperatorTok{[}\NormalTok{\_}\OperatorTok{],}\NormalTok{ prime}\OperatorTok{[}\NormalTok{\_}\OperatorTok{],}\NormalTok{ tot}\OperatorTok{,}\NormalTok{ d}\OperatorTok{[}\NormalTok{\_}\OperatorTok{],}\NormalTok{ x}\OperatorTok{;}
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  Package hyperref Warning: Rerun to get /PageLabels entry.
  LaTeX Warning: Label(s) may have changed. Rerun to get cross-references right.
[INFO] [makePDF] LaTeX run number 2
[INFO] [makePDF] LaTeX output
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   restricted \write18 enabled.
  entering extended mode
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  LaTeX Font Warning: Font shape `TU/ARPLUKaiCN(0)/b/n' undefined
  (Font)              using `TU/ARPLUKaiCN(0)/m/n' instead on input line 117.
  
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  Package xeCJK Warning: Unknown CJK family `\CJKttdefault' is being ignored.
  (xeCJK)                
  (xeCJK)                Try to use `\setCJKmonofont[<...>]{<...>}' to define
  (xeCJK)                it.
  
  [2] (.../input.aux)
  
  LaTeX Font Warning: Some font shapes were not available, defaults substituted.
  
   )
  Output written on .../input.pdf (2 pages).
  Transcript written on .../input.log.