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  "body": "\n## Definition\n\nLet $X$ be a set, let $f \\colon X \\to X$ be a map, and let $x \\in X$ be a [[periodic point|T75CZ5H47N]] of $f$.\nThe *minimal period*, or *prime period*, of $x$ is the least integer $n \\ge 1$ such that $f^n(x) = x$, where $f^n$ is the $n$-th [[iterate|AYVTR2ES5B]] of $f$.\n\n## Remarks\n\nLet $p$ be the minimal period of $x$.\nThen $f^m(x) = x$ holds exactly when $p$ divides $m$: write $m = qp + r$ with $0 \\le r \u003c p$; then $x = f^m(x) = f^r(x)$, so $r = 0$ by minimality.\nThe points $x, f(x), \\dots, f^{p-1}(x)$ are distinct, so the orbit of $x$ has exactly $p$ points.\nA point has minimal period $1$ exactly when it is a fixed point.\n\nExample: for $f(x) = -x$ on $\\R$ and $x = 1$, one has $f(1) = -1 \\ne 1$ and $f^2(1) = 1$, so the minimal period is $2$.\n\nNon-example: for the same map, $2$ is not the minimal period of $0$.\nIt is a period, since $f^2(0) = 0$, but already $f(0) = 0$, so the minimal period of $0$ is $1$.\n\nIn Lean's Mathlib, `Function.minimalPeriod f x` is defined for all $x$ and takes the value $0$ when $x$ is not periodic.\n",
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