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  "body": "\n## Statement\n\nLet $n$ be a positive integer, and let $F_m$ be the [[Fibonacci numbers|0GA9K57A39]].\nThen\n$$F_{n-1} F_{n+1} - F_n^2 = (-1)^n.$$\n\n## Remarks\n\nExamples: for $n = 5$, $F_4 F_6 - F_5^2 = 3 \\cdot 8 - 25 = -1$. For $n = 6$, $F_5 F_7 - F_6^2 = 5 \\cdot 13 - 64 = 1$.\nA short proof takes determinants in the matrix identity $\\begin{pmatrix} 1 \u0026 1 \\\\ 1 \u0026 0 \\end{pmatrix}^n = \\begin{pmatrix} F_{n+1} \u0026 F_n \\\\ F_n \u0026 F_{n-1} \\end{pmatrix}$.\nCatalan's identity $F_n^2 - F_{n-r} F_{n+r} = (-1)^{n-r} F_r^2$ generalizes it. The case $r = 1$ is Cassini's identity.\n\n## Explanations\n\n### L1 human\n\nThe product of the two neighbours of a Fibonacci number differs from its square by exactly $1$.\nThe sign alternates: $1 \\cdot 3 - 2^2 = -1$, then $2 \\cdot 5 - 3^2 = 1$, then $3 \\cdot 8 - 5^2 = -1$.\n",
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    "name": "Cassini's identity",
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  "msc": [
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  "name": "Cassini's identity",
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