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  "body": "\n## Summary\n\nThe periodicity equation is the [[functional equation|SRAN4EKWS0]] $f(x + p) = f(x)$ for a fixed nonzero number $p$ and an unknown function $f$ on $\\R$.\nIts solutions are exactly the [[periodic functions|0ATF6Q0633]] with period $p$.\nIt is the simplest linear difference equation with constant coefficients, and many other equations reduce to it.\nFor example, the general solution of $g(x + 1) = g(x) + 1$ is $g(x) = x + f(x)$, with $f$ a solution of the periodicity equation for $p = 1$.\n\n## Explanations\n\n### L1 human\n\nThe equation says that the function repeats itself after a shift by $p$.\nA solution is fixed by its values on one interval of length $p$, and these values can be chosen freely.\nFor example, $\\sin x$ solves it for $p = 2\\pi$.\nThe fractional part $x - \\floor{x}$ solves it for $p = 1$.\n",
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