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  "body": "\n## Definition\n\nLet $n$ be a positive odd integer with [[prime factorization|0AEKDFCYFA]] $n = p_1^{e_1} \\cdots p_k^{e_k}$, and let $a$ be an integer.\nThe Jacobi symbol is\n$$\\left(\\frac{a}{n}\\right) = \\left(\\frac{a}{p_1}\\right)^{e_1} \\cdots \\left(\\frac{a}{p_k}\\right)^{e_k},$$\nwhere each factor on the right is a [[Legendre symbol|01J8AMV9RS]].\nFor $n = 1$, the empty product gives $\\left(\\frac{a}{1}\\right) = 1$.\n\n## Remarks\n\nFor a prime $n$, the Jacobi symbol equals the Legendre symbol.\nIf $\\left(\\frac{a}{n}\\right) = -1$, then $a$ is not a quadratic residue modulo $n$.\nThe converse fails: $\\left(\\frac{2}{15}\\right) = \\left(\\frac{2}{3}\\right) \\left(\\frac{2}{5}\\right) = 1$, but $2$ is not a square modulo $15$.\nThe symbol is $0$ exactly when $\\gcd(a, n) \u003e 1$.\n\n## Explanations\n\n### L1 human\n\nThe Jacobi symbol extends the Legendre symbol from odd primes to all odd positive integers.\nIt multiplies the Legendre symbols for the prime factors of $n$.\nIts main use is fast computation, because it obeys [[quadratic reciprocity|0YY06KP3ZQ]] without the need to factor numbers.\n",
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