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  "body": "\n## Statement\n\nLet $n$ be a positive integer.\nThen $n = x^2 + y^2$ for some integers $x$ and $y$ if and only if $\\nu_p(n)$ is even for each prime $p \\equiv 3 \\pmod 4$.\nHere $\\nu_p$ is the [[p-adic valuation|04VCPS30ZE]].\n\n## Remarks\n\nFor example, $45 = 3^2 \\cdot 5 = 6^2 + 3^2$, but $21 = 3 \\cdot 7$ is not a sum of two squares.\nThe \"if\" part follows from [[Fermat's theorem on sums of two squares|043B3ENSTY]] and the [[Brahmagupta–Fibonacci identity|0K2DMACEZ4]].\nZero squares are allowed, so each square and each double of a square is a sum of two squares.\n\n## Explanations\n\n### L1 human\n\nTo decide whether $n$ is a sum of two squares, factor $n$ into primes.\nLook only at the primes that leave remainder $3$ on division by $4$. Each of them must appear an even number of times.\n",
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