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  "body": "\n## Statement\n\nLet $m$ and $n$ be distinct natural numbers.\nThen the [[Fermat numbers|0R03B54V4Z]] $F_m$ and $F_n$ are [[coprime|0GFKP8F8AG]].\n\n## Remarks\n\nThe proof uses $F_0 F_1 \\cdots F_{n-1} = F_n - 2$. A common divisor of $F_m$ and $F_n$ with $m \u003c n$ divides $2$. Fermat numbers are odd, so the divisor is $1$.\nThe result is often called Goldbach's theorem, after Christian Goldbach.\nA consequence is a new proof that there are [[infinitely many primes|0292EPPPT0]]: each $F_n$ has a prime factor, and these primes are all distinct.\n\n## Explanations\n\n### L1 human\n\nNo two Fermat numbers share a prime factor.\nFor example, $F_2 = 17$ and $F_3 = 257$ are coprime, and $F_5 = 641 \\cdot 6700417$ has no factor in common with $3$, $5$, $17$, $257$ or $65537$.\n",
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