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  "body": "\n## Proof\n\nWe reduce to the equation $x^4 - y^4 = z^2$.\n\n1. Suppose $a$, $b$ and $c$ are positive integers with $a^4 + b^4 = c^4$.\n2. Then $c^4 - b^4 = a^4 = (a^2)^2$.\n3. So $x = c$, $y = b$ and $z = a^2$ are positive integers with $x^4 - y^4 = z^2$.\n4. This contradicts [[Fermat's right triangle theorem|086BE0J3FZ]], which says that $x^4 - y^4 = z^2$ has no solution in positive integers.\n5. Hence there are no positive integers $a$, $b$ and $c$ with $a^4 + b^4 = c^4$.\n\n## Idea\n\nA fourth power is a square. So a solution of $a^4 + b^4 = c^4$ is a solution of the stronger equation $c^4 - b^4 = z^2$ with $z = a^2$. Fermat's descent rules that out.\n",
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