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  "body": "\n## Statement\n\nThere are no positive integers $a$, $b$ and $c$ with $a^3 + b^3 = c^3$.\n\n## Remarks\n\nA sum of two positive cubes is never a cube.\nStandard proofs use [[infinite descent|0EFYCFHYFG]] in the ring $\\mathbb{Z}[\\omega]$ of Eisenstein integers, where $\\omega$ is a primitive cube root of unity.\nThis case is part of [[Fermat's Last Theorem|0Y5X3SDBCZ]].\n\n## History\n\nFermat mentioned the case $n = 3$ in letters of 1636, 1640 and 1657.\nEuler published a proof in 1770. Many other proofs followed, for example by Legendre.\n\n## Explanations\n\n### L1 human\n\nYou can never write a cube as a sum of two positive cubes.\nFor example, $6^3 = 216$, and $3^3 + 4^3 + 5^3 = 216$ needs three cubes, not two.\n",
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