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  "body": "\n## Statement\n\nLet $p$ be a [[prime number|0SN0MAEKYX]].\nThere is an integer $x$ with $x^2 \\equiv -1 \\pmod p$ if and only if $p \\not\\equiv 3 \\pmod 4$.\n\n## Remarks\n\nFor $p = 2$, take $x = 1$.\nFor odd $p$, the statement is the [[first supplement to quadratic reciprocity|0PHBS78HKT]].\nFor $p \\equiv 1 \\pmod 4$, an explicit solution is $x = \\left(\\frac{p-1}{2}\\right)!$, by [[Wilson's theorem|03KGGKTB2E]].\nThe statement is a step in the proof of [[Fermat's theorem on sums of two squares|043B3ENSTY]].\n\n## Explanations\n\n### L1 human\n\nThe primes $2$, $5$, $13$ and $17$ have a square root of $-1$: for example $5^2 = 25 \\equiv -1 \\pmod{13}$.\nThe primes $3$, $7$ and $11$ have none.\nThe rule is that $p$ must not leave remainder $3$ on division by $4$.\n",
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    "name": "−1 is a square modulo p if and only if p is not 3 modulo 4",
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        "locator": "q = ±1 and the first supplement",
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  "msc": [
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  "name": "−1 is a square modulo p if and only if p is not 3 modulo 4",
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