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  "body": "\n## Statement\n\nLet $p$ be an odd [[prime number|0SN0MAEKYX]].\nThen there are integers $x$ and $y$ with $p = x^2 + y^2$ if and only if $p \\equiv 1 \\pmod 4$.\n\n## Remarks\n\nThe prime $2 = 1^2 + 1^2$ is also a sum of two squares. So a prime $p$ is a sum of two squares if and only if $p \\not\\equiv 3 \\pmod 4$.\nFor example, $5 = 1 + 4$, $13 = 4 + 9$ and $29 = 4 + 25$. For $p \\equiv 1 \\pmod 4$, the representation is unique up to order and signs.\nThe \"only if\" part is easy: a square is $0$ or $1$ modulo $4$, so $x^2 + y^2 \\not\\equiv 3 \\pmod 4$.\nOne proof of the other part starts from a [[square root of $-1$ modulo $p$|00DW862T78]] and uses [[Gaussian integers|0B268GVX95]] or [[Minkowski's theorem|0K387S2PKQ]].\n\n## History\n\nAlbert Girard stated the result in a book published in 1625.\nFermat stated it in a letter to Mersenne of 25 December 1640, so it is also called Fermat's Christmas theorem.\nEuler gave the first proof, by infinite descent, in 1749.\n\n## Explanations\n\n### L1 human\n\nThe primes $5, 13, 17, 29$ leave remainder $1$ on division by $4$, and each of them is a sum of two squares.\nThe primes $3, 7, 11, 19$ leave remainder $3$, and none of them is a sum of two squares.\n",
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