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  "body": "\n## Statement\n\nThere are no positive integers $a$, $b$ and $c$ with $a^4 + b^4 = c^4$.\n\n## Remarks\n\nA stronger statement holds: $x^4 + y^4 = z^2$ has no solution in positive integers.\nThe theorem also follows from [[Fermat's right triangle theorem|086BE0J3FZ]], in the form that $x^4 - y^4 = z^2$ has no solution in positive integers.\nThe proof uses [[infinite descent|0EFYCFHYFG]] and the [[classification of Pythagorean triples|0KF3J2KAER]].\nSince $4$ divides every power of $2$ above $2$, the case $n = 4$ settles [[Fermat's Last Theorem|0Y5X3SDBCZ]] for all such exponents.\n\n## History\n\nFermat's proof of his right triangle theorem, by infinite descent, gives this case.\n\n## Explanations\n\n### L1 human\n\nThe equation $a^4 + b^4 = c^4$ has no whole-number solutions.\nThe proof assumes a solution and builds a smaller one from it, which leads to a contradiction.\n",
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