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  "body": "\n## Definition\n\nLet $p$ be an odd [[prime number|0SN0MAEKYX]] and let $a$ be an integer.\nThe Legendre symbol $\\left(\\frac{a}{p}\\right)$ is the number\n$$\\left(\\frac{a}{p}\\right) = \\begin{cases} 0 \u0026 \\text{if } p \\mid a, \\\\ 1 \u0026 \\text{if } p \\nmid a \\text{ and } a \\text{ is a quadratic residue modulo } p, \\\\ -1 \u0026 \\text{if } a \\text{ is a quadratic nonresidue modulo } p. \\end{cases}$$\nHere quadratic residue has the meaning of the [[definition of a quadratic residue|0SQDB87Z3T]].\n\n## Remarks\n\nThe value depends only on the class of $a$ modulo $p$.\nFor example, $\\left(\\frac{2}{7}\\right) = 1$ because $3^2 \\equiv 2 \\pmod 7$, and $\\left(\\frac{3}{7}\\right) = -1$.\n[[Euler's criterion|0M7DEDKSYB]] gives the value as $a^{(p-1)/2}$ modulo $p$.\nThe [[law of quadratic reciprocity|085MJS6907]] helps to compute the symbol quickly.\n\n## Explanations\n\n### L1 human\n\nThe Legendre symbol is a short way to say whether $a$ is a square modulo $p$.\nThe value $1$ means yes, the value $-1$ means no, and the value $0$ means that $a$ is $0$ modulo $p$.\n",
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