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  "body": "\n## Statement\n\nLet $p$ be a [[prime number|0SN0MAEKYX]].\nThen $p$ is a [[Gaussian prime|0GTEHJQRFK]] if and only if $p \\equiv 3 \\pmod 4$.\n\n## Remarks\n\nIf $p = 2$, then $2 = (1 + i)(1 - i)$, and $1 + i$ is a Gaussian prime.\nIf $p \\equiv 1 \\pmod 4$, then $p = a^2 + b^2 = (a + bi)(a - bi)$ by [[Fermat's theorem on sums of two squares|043B3ENSTY]], and $a \\pm bi$ are Gaussian primes.\nIf $p \\equiv 3 \\pmod 4$, then $p$ is not a sum of two squares. So no Gaussian integer has norm $p$, and $p$ cannot split.\nFor example, $3$ and $7$ are Gaussian primes, but $13 = (3 + 2i)(3 - 2i)$.\n\n## Explanations\n\n### L1 human\n\nAn ordinary prime stays prime among the Gaussian integers exactly when it leaves remainder $3$ on division by $4$.\nThe other primes factor, because they are sums of two squares.\n",
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