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  "body": "\n## Statement\n\nLet $m$ and $n$ be [[coprime|0GFKP8F8AG]] positive integers with $m \u003e n$ and $m - n$ odd.\nThen $(m^2 - n^2, \\, 2mn, \\, m^2 + n^2)$ is a primitive [[Pythagorean triple|01G1WY1GVX]].\nConversely, each primitive Pythagorean triple $(a, b, c)$ with $b$ even arises in this way from exactly one such pair $(m, n)$.\nEach Pythagorean triple is $(ka, kb, kc)$ for a positive integer $k$ and a primitive triple $(a, b, c)$, up to the order of the first two entries.\n\n## Remarks\n\nThe formula is called Euclid's formula.\nFor example, $(m, n) = (2, 1)$ gives $(3, 4, 5)$, and $(m, n) = (3, 2)$ gives $(5, 12, 13)$.\nOne proof writes $b^2 = (c - a)(c + a)$ and uses unique factorization. Another uses the rational points on the unit circle.\n\n## Explanations\n\n### L1 human\n\nEuclid's formula produces every right triangle with whole-number sides, from two numbers $m$ and $n$.\nThe identity $(m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2$ shows that the formula always works.\n",
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