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## Definition: Jacobi Symbol

Let $n > 0$ be an odd and positive integer with the factorization $n=p_1^{e_1}\cdots p_r^{e_r}.$ For an integer $a\in\mathbb Z,$ the Jacobi symbol of $a$ modulo $n$ is an arithmetic function defined by

$$\left(\frac an\right):=\left(\frac a{p_1}\right)^{e_1}\cdots \left(\frac a{p_r}\right)^{e_r},$$
where $\left(\frac a{p_i}\right)$ denote the Legendre symbols of $a$ modulo the prime numbers $p_i$ dividing $n.$

| | | | | created: 2019-06-15 08:47:06 | modified: 2019-06-15 08:51:59 | by: bookofproofs | references: [1272], [8187]