# 雅可比符号

## 定义

 ${\displaystyle \left({\frac {a}{p}}\right)={\begin{cases}\;\;\,0\\+1\\-1\end{cases}}}$ 如果p整除a； 如果存在整数 ${\displaystyle X}$ 使得 ${\displaystyle X^{2}\equiv a{\pmod {p}}}$ 且p不整除a 如果不存在整数 ${\displaystyle X}$ 使得 ${\displaystyle X^{2}\equiv a{\pmod {p}}}$
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${\displaystyle ({\frac {a}{p}})=1}$ 时，稱${\displaystyle a}$ 是模${\displaystyle p}$的二次剩餘；当${\displaystyle ({\frac {a}{p}})=-1}$ 时，稱${\displaystyle a}$ 是模${\displaystyle p}$的二次非剩餘。

${\displaystyle m}$ 是一个正奇数，其质因数分解式为 ${\displaystyle m=\prod _{i=1}^{s}p_{i}}$，并且正整数 ${\displaystyle a}$ 满足 ${\displaystyle (m,a)=1}$ 那么定义${\displaystyle ({\frac {a}{m}})=\prod _{i=1}^{s}({\frac {a}{p_{i}}})}$

## 注释

1. ^ C.G.J.Jacobi "Uber die Kreisteilung und ihre Anwendung auf die Zahlentheorie", Bericht Ak. Wiss. Berlin (1837) pp 127-136

## 参考来源

• Ireland, Kenneth; Rosen, Michael, A Classical Introduction to Modern Number Theory (Second edition), New York: Springer, 1990, ISBN 0-387-97329-X
• Gauss, Carl Friedrich; Maser, H. (translator into German), Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition), New York: Chelsea, 1965, ISBN 0-8284-0191-8
• Gauss, Carl Friedrich; Clarke, Arthur A. (translator into English), Disquisitiones Arithemeticae (Second, corrected edition), New York: Springer, 1986, ISBN 0387962549