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  "body": "\n## Statement\n\nLet $n$ be a positive integer and let $F_n = 2^{2^n} + 1$ be the [[Fermat number|0R03B54V4Z]].\nThen $F_n$ is prime if and only if\n$$3^{(F_n - 1)/2} \\equiv -1 \\pmod{F_n}.$$\n\n## Remarks\n\nFor example, $F_1 = 5$ and $3^2 = 9 \\equiv -1 \\pmod 5$.\nThe power can be computed by $2^n - 1$ squarings with [[modular exponentiation|04DR5PM25N]].\nThe \"only if\" part follows from [[Euler's criterion|0M7DEDKSYB]] and quadratic reciprocity, since $3$ is a nonresidue modulo $F_n$.\n\n## History\n\nThe test is named after Théophile Pépin.\n\n## Explanations\n\n### L1 human\n\nPépin's test decides whether a Fermat number is prime with one large modular power.\nThe Fermat numbers grow so fast that the test is practical only for small $n$.\n",
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