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  "body": "\n## Statement\n\nThere is no right triangle with integer side lengths whose area is the square of an integer.\nEquivalently, the equation $x^4 - y^4 = z^2$ has no solution in positive integers $x$, $y$ and $z$.\n\n## Remarks\n\nIn terms of a [[Pythagorean triple|01G1WY1GVX]] $(a, b, c)$, the theorem says that $ab/2$ is never a perfect square.\nA consequence is [[Fermat's Last Theorem for n = 4|0E741HSMJ1]], since $a^4 + b^4 = c^4$ gives $c^4 - b^4 = (a^2)^2$.\nThe proof is by [[infinite descent|0EFYCFHYFG]].\n\n## History\n\nThe proof was published in 1670 among the works of Pierre de Fermat, after his death.\nIt is the only complete proof by Fermat that survives.\n\n## Explanations\n\n### L1 human\n\nThe triangle with sides $3$, $4$, $5$ has area $6$, and the one with sides $5$, $12$, $13$ has area $30$.\nFermat proved that such an area can never be a perfect square.\n",
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