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  "body": "\n## Statement\n\nLet $(F_n)_{n \\ge 0}$ be the Fibonacci sequence, given by $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for $n \\ge 0$.\nThen $F_n$ and $F_{n+1}$ are [[coprime|0GFKP8F8AG]] for each $n \\ge 0$, that is, $\\gcd(F_n, F_{n+1}) = 1$.\n\n## Remarks\n\nA proof by induction: $\\gcd(F_{n+1}, F_{n+2}) = \\gcd(F_{n+1}, F_{n+2} - F_{n+1}) = \\gcd(F_{n+1}, F_n)$, and $\\gcd(F_0, F_1) = 1$.\nA stronger fact is $\\gcd(F_m, F_n) = F_{\\gcd(m, n)}$ for all $m, n \\ge 0$.\nThe [[Euclidean algorithm|0TRNT4X7J0]] on $F_{n+1}$ and $F_n$ takes many steps, and consecutive Fibonacci numbers give its worst case.\n\n## Explanations\n\n### L1 human\n\nTwo neighbours in the Fibonacci sequence $0, 1, 1, 2, 3, 5, 8, 13, \\dots$ never have a common factor larger than $1$.\nA common factor of two neighbours would also divide the term before them, and so on down to $F_1 = 1$.\n",
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