初等数论结论

说明：本文只有定理、结论的记录。如果想看证明的话可以看我的初等数论笔记
由于本人（饱受学校压榨）时间精力有限，更新可能比较慢且不定期（尤其是证明）

整除

带余除法

\(\underline{\textbf{Theorem 1}}\quad\forall n\in\mathbb{Z},\,m\in\mathbb{Z^+},\,\exists\,\)唯一\(\,q,r\in\mathbb{Z},\,0\le r<m,\,\text{s.t.}\,n=qm+r.\)

裴蜀定理

\(\underline{\textbf{Theorem 2}}\quad a,b\in\mathbb{Z},\,\)且\(\,\gcd(a,b)=d,\,\)那么\(\,\forall x,y\in\mathbb{Z},\,d\mid ax+by.\)
特别地\(,\,\exists x,y\in\mathbb{Z},\,\text{s.t.}\,ax+by=d.\)

算术基本定理

\(\underline{\textbf{Theorem 3}}\quad\forall n\in\mathbb{Z^+},\,n>1,\,\exists\,\)质数\(\,p_1<p_2<\ldots<p_s\,\)和正整数\(\,\alpha_1,\alpha_2,\ldots,\alpha_s,\,\text{s.t.}\,n=\prod\limits_{i=1}^{s}p_i^{\alpha_i}.\,\)且这种表示是唯一的\(.\)

两个数论函数

\(\underline{\textbf{Definition}}\quad\tau(n)\,\)定义为\(\,n\,\)的正因子个数\(.\,\sigma(n)\,\)定义为\(\,n\,\)的所有正因子之和\(.\)
\(\underline{\textbf{Theorem 4}}\quad\)容易得到\(\,\tau(n)=\sum\limits_{d\mid n}1=\prod\limits_{i=1}^s(1+\alpha_i).\)
\(\,\sigma(n)=\sum\limits_{d\mid n}d=\sum\limits_{ \begin{matrix} (\beta_1,\ldots,\beta_s)\\ 0\le \beta_i\le\alpha_i \end{matrix}} p_1^{\beta_1}\ldots p_s^{\beta_s}=\prod\limits_{i=1}^s\sum\limits_{j=0}^{\alpha_i}p_i^j=\prod\limits_{i=1}^s\frac{p_i^{\alpha_i+1}-1}{p_i-1}.\)
且\(\,\tau(n),\,\sigma(n)\,\)均为积性函数\(.\)

\(\underline{\textbf{Remark}}\quad\)算术函数与积性函数
定义在\(\,\mathbb{Z^+}\,\)上的函数称为算术函数\(.\)
设\(\,f(n)\,\)为算术函数\(.\,\)若\(\,f(n)\,\)不恒为\(\,0,\,\)且\(\,\forall m,n\in\mathbb{Z^+}\,\)满足\(\,\gcd(m,n)=1,\,\)均有\(\,f(mn)=f(m)f(n),\,\)则称\(\,f(n)\,\)为积性函数\(.\,\)特别地\(,\,\)若\(\,\forall m,n\in\mathbb{Z^+},\,f(mn)=f(m)f(n),\,\)则称其为完全积性函数\(.\)

完全数

\(\underline{\textbf{Definition}}\quad\)若\(\,n\in\mathbb{Z^+},\,\sigma(n)=2n,\,\)则称\(\,n\,\)为完全数\(.\)
\(\underline{\textbf{Theorem 5}}\)
\((1)\,n\,\)为偶完全数\(\,\Longleftrightarrow n=2^{p-1}(2^p-1).\,\)其中\(\,p,2^p-1\,\)均为素数\(.\)
\((2)\,\)若\(\,n\,\)为奇完全数\(,\,\)则\(\,n=p^sm^2.\,\)其中\(\,p\,\)为奇质数满足\(\,p\equiv s\equiv1\pmod4,\,\)且\(\,p\nmid m.\)

\(\underline{\textbf{Remark}}\quad\)
(1)有如下猜想:
\(\quad\)①奇完全数不存在\(.\)
\(\quad\)②偶完全数有无穷多个\(.\)
\(\quad\)③\(\,\exists\,\)无穷多质数\(\,p,\,\text{s.t.}\,2^p-1\,\)为质数\(.\)
\(\quad\)其中由上面的结论知②与③等价\(.\)
(2)\(\,p\,\)为任一质数\(,\,\)称\(\,M_p=2^p-1\,\)为梅森\((Mersenne)\)数\(.\,\)则③即为梅森素数猜想\(.\)

同余

欧拉\((Euler)\)定理

欧拉函数

\(\underline{\textbf{Definition}}\quad\varphi(n)\,\)定义为\(\,1,2,\ldots,n\,\)中与\(\,n\,\)互质的数的个数\(.\)
设\(\,n=\prod\limits_{i=1}\limits^{s}p_i^{\alpha_i},\,\)其中\(\,p_1<p_2<\ldots<p_s\,\)为质数\(,\,\alpha_i\in\mathbb{Z_+},\,i=1,2,...,s.\)
则\(\,\varphi(n)=n\prod\limits_{i=1}^{s}\left(1-\frac{1}{p_i} \right).\,\)特别地\(,\,\varphi(1)=1,\,\varphi(p)=p-1,\,\)其中\(\,p\,\)为质数\(.\)

欧拉定理

\(\underline{\textbf{Theorem 6}}\quad\)若\(\,a,m\in\mathbb{Z^+},\,\gcd(a,m)=1,\,\)则有\(\,a^{\varphi(m)}\equiv1\pmod m.\)

费马\((Fermat)\)小定理

\(\underline{\textbf{Corollary 7}}\quad\)若\(\,a\in\mathbb{Z^+},\,p\,\)为质数\(,\,\gcd(a,p)=1,\,\)则有\(\,a^p\equiv a\pmod p,\,\)即\(\,a^{p-1}\equiv1\pmod p.\)

威尔逊\((Wilson)\)定理

\(\underline{\textbf{Theorem 8}}\quad p\,\)为质数\(\,\Longleftrightarrow(p-1)!\equiv-1\pmod p.\)

推广

\(\underline{\textbf{Theorem 9}}\quad(n-1)!\equiv\begin{cases} -1,&n\,为质数\\ 2,&n=4\\ 0,&n=1\,或\,n\,为大于\,4\,的合数\\ \end{cases}\pmod n.\)

中国剩余定理\((CRT)\)

\(\underline{\textbf{Theorem 10}}\quad m_1,m_2, ... ,m_s\in\mathbb{Z}\,\)满足\(\,\forall 1\le i<j\le s,\,\gcd(m_i,m_j)=1.\,\forall a_1,a_2, ... ,a_s\in\mathbb{Z},\,\)方程组
\(\begin{equation}\label{test}\left\{\begin{aligned}x&\equiv a_1\pmod {m_1}\\ x&\equiv a_2\pmod {m_2}\\ &\cdots \\ x&\equiv a_s\pmod {m_s}\\\end{aligned}\right.\end{equation}\)
有整数解\(.\,\)且若\(\,x,x'\,\)均为\(\,(\ref{test})\,\)的解\(,\,\)则有\(\,x\equiv x'\pmod M,\,\)其中\(\,M=\prod\limits_{i=1}\limits^s m_i.\)
具体的\(,\,x\equiv\sum\limits_{i=1}^{s}a_i\times\frac{M}{m_i}\times\left[\left( \frac{M}{m_i} \right)^{-1} \right]{m_i}\pmod M.\)
其中\(\,\left[\left( \frac{M}{m_i} \right)^{-1} \right]{m_i}\,\)为\(\,\frac{M}{m_i}\,\)在\(\bmod m_i\,\)意义下的逆元\(,\,\)即\(\,\frac{M}{m_i}\times\left[\left( \frac{M}{m_i} \right)^{-1} \right]{m_i}\equiv1\pmod {m_i}.\)

维诺格拉多夫\((Vinogradov)\)引理

\(\underline{\textbf{Lemma 11}}\quad a,b,n\in\mathbb{Z^+},\,n>1,\,\gcd(ab,n)=1,\,\alpha\in\mathbb{R^+},\,\)则\(\,\exists x,y\in\mathbb{Z},\,x,y\,\)不全为\(\,0,\,\)且\(\,\lvert x \rvert\le\alpha,\,\lvert y\rvert\le\frac{n}{\alpha},\)\(\,\text{s.t.}\,ax\equiv by\pmod n.\)

关于\(v_p(n)\)

定义

\(\underline{\textbf{Definition}}\quad p\,\)为任一质数\(,\,n\in\mathbb{Z},\,\)若\(\,p^k\mid\mid n,\,\)则\(\,v_p(n)=k.\)

勒让德\((Legendre)\)公式

\(\underline{\textbf{Theorem 12}}\quad v_p(n!)=\sum\limits_{k=1}^{\infty}\left[\frac{n}{p^k}\right].\)

\(p\,\text-\,\)进制形式

\(\underline{\textbf{Theorem 13}}\quad n\,\)的\(\,p\,\text{-}\,\)进制表示为\(\sum\limits_{i=0}^{t}a_ip^i.\,\)其中\(\,a_i\in\mathbb{Z},\,0\le a_i\le p-1,\,a_t>0.\,\)记\(\,S(n)=\sum\limits_{i=0}^{t}a_i.\,\)则\(\,v_p(n)=\frac{n-S(n)}{p-1}.\)
上述公式可以导出如下结论:
\(\underline{\textbf{Corollary 14}}\quad p\,\)为质数\(,\,n,k\in\mathbb{N},\,0\le k\le n,\,n\,\)的\(\,p\,\text{-}\,\)进制表示为\(\sum\limits_{i=0}^{t}a_ip^i,\,k\,\)的\(\,p\,\text{-}\,\)进制表示为\(\sum\limits_{i=0}^{t}b_ip^i.\,\)
其中\(\,a_i,b_i\in\mathbb{Z},\,0\le a_i,b_i\le p-1,\,a_t>0.\,\)
则\(\,p\nmid \mathrm{C}_ n^k\Longleftrightarrow\forall 0\le i\le t,\,\)均有\(\,b_i\le a_i.\)
此即卢卡斯定理的推论.

\(\underline{\textbf{Remark}}\quad\)卢卡斯\((Lucas)\)定理
\(\underline{\textbf{Theorem 15}}\quad p\,\)为质数\(,\,n,k\in\mathbb{N},\,0\le k\le n,\,n\,\)的\(\,p\,\text{-}\,\)进制表示为\(\sum\limits_{i=0}^{t}a_ip^i,\,k\,\)的\(\,p\,\text{-}\,\)进制表示为\(\sum\limits_{i=0}^{t}b_ip^i.\,\)
其中\(\,a_i,b_i\in\mathbb{Z},\,0\le a_i,b_i\le p-1,\,a_t>0.\,\)
则\(\,\mathrm{C}_ n^k\equiv\prod\limits_{i=0}^{t}\mathrm{C}_ {a_i}^{b_i}\pmod p.\)

升幂定理\((LTE)\)

\(\underline{\textbf{Theorem 16}}\quad n\in\mathbb{Z^+},\,x,y\in\mathbb Z,\,\)奇质数\(\,p\,\)满足\(\,\gcd(p,xy)=1,\,p\mid x-y,\)
则有\(\,v_p(x^n-y^n)=v_p(x-y)+v_p(n).\)
\(\underline{\textbf{Corollary 17}}\quad n\,\)为奇数时\(,\,v_p(x^n+y^n)=v_p(x+y)+v_p(n).\)
\(\underline{\textbf{Theorem 18}}\quad n\in\mathbb{Z^+},\,x,y\in\mathbb Z,\,\)满足\(\,\gcd(2,xy)=1,\,\)则\(\,2\mid x-y,\)
于是有\(\,v_2(x^n-y^n)=\begin{cases} v_2(x-y),&n\,为奇数\\ v_2(x-y)+v_2(x+y)+v_2(n)-1,&n\,为偶数\\ \end{cases}\ .\)

二次剩余

以下\(\,p\,\)均表示任一奇素数.

定义

\(\underline{\textbf{Definition}}\quad m,n\in\mathbb{Z^+},\,\)若\(\,\exists x\in\mathbb{Z^+},\,\text{s.t.}\,x^2\equiv n\pmod m,\,\)则称\(\,n\,\)为模\(\,m\,\)的二次剩余\(.\,\)否则称为二次非剩余\(.\)

性质

\(\underline{\textbf{Property 19}}\quad\)模\(\,p\,\)的二次剩余和二次非剩余各有\(\,\frac{p-1}{2}\,\)个\(.\)
\(\underline{\textbf{Property 20}}\quad\)任意一个模\(\,p\,\)的二次剩余恰与如下序列中的一个数模\(\,p\,\)同余:\(\,1^2,2^2,\ldots,(\frac{p-1}{2})^2.\)
\(\underline{\textbf{Property 21}}\quad\)若\(\,d\,\)为模\(\,p\,\)的二次剩余\(,\,\)则\(\,x^2\equiv d\pmod p\,\)的解的个数为\(\,2.\)
\(\underline{\textbf{Property 22}}\quad a\,\)为模\(\,p\,\)的二次剩余,则\(\,a\cdot1^2,a\cdot2^2,\ldots,a\cdot(\frac{p-1}{2})^2\,\)恰为模\(\,p\,\)的所有二次剩余\(.\)\(b\,\)为模\(\,p\,\)的二次非剩余,则\(\,b\cdot1^2,b\cdot2^2,\ldots,b\cdot(\frac{p-1}{2})^2\,\)恰为模\(\,p\,\)的所有二次非剩余\(.\)

勒让德符号

\(\underline{\textbf{Definition}}\quad n\in\mathbb{Z},\,\)则\(\,Legendre\,\)符号为\(\,\left(\dfrac{n}{p}\right)=\begin{cases} 1,&n\,为模\,p\,的二次剩余\\ -1,&n\,为模\,p\,的二次非剩余\\ 0,&n\,为\,p\,的倍数\\ \end{cases}.\)

欧拉判别法

\(\underline{\textbf{Theorem 23}}\quad a\in\mathbb{Z^+},\,\gcd(a,p)=1,\,\)则
\((1)\,a\,\)为模\(\,p\,\)的二次剩余\(\,\Longleftrightarrow a^{\frac{p-1}{2}}\equiv1\pmod p.\)
\((2)\,a\,\)为模\(\,p\,\)的二次非剩余\(\,\Longleftrightarrow a^{\frac{p-1}{2}}\equiv-1\pmod p.\)
用\(\,Legendre\,\)符号表示即为\(\,\left(\dfrac{a}{p}\right)=a^{\frac{p-1}{2}}.\)

高斯二次互反律

\(\underline{\textbf{Theorem 24}}\quad\)
\((1)\,\left(\dfrac{-1}{p}\right)=(-1)^{\frac{p-1}{2}}.\)
\((2)\,\left(\dfrac{2}{p}\right)=(-1)^{\frac{p^2-1}{8}}.\)
\((3)\,p,q\,\)为奇素数\(,\,\)则\(\,\left(\dfrac{q}{p}\right)\cdot\left(\dfrac{p}{q}\right)=(-1)^{\frac{p-1}{2}\cdot\frac{q-1}{2}}.\)