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  "body": "\n## Summary\n\nThe Miller–Rabin test is a probabilistic [[primality test|0EGJZHZ02C]] for an odd integer $n \u003e 2$.\nWrite $n - 1 = 2^s d$ with $d$ odd, and choose a base $a$ with $1 \u003c a \u003c n - 1$.\nThe number $n$ passes for base $a$ if $a^d \\equiv 1 \\pmod n$, or if $a^{2^r d} \\equiv -1 \\pmod n$ for some $r$ with $0 \\le r \u003c s$.\nEach odd prime $n$ passes for each base. If $n$ fails for some base, then $n$ is composite.\n\n## Remarks\n\nThe prime case follows from [[Fermat's little theorem|0M25PRN3WQ]] and the fact that $\\pm 1$ are the only square roots of $1$ modulo a prime.\nIf $n$ is an odd composite number, then at most $\\frac{1}{4}$ of the bases let it pass. So $k$ random bases give an error probability of at most $4^{-k}$.\nGary Miller found the test in 1976, in a deterministic form that depends on the extended Riemann hypothesis. Michael Rabin made it an unconditional probabilistic test in 1980.\n\n## Explanations\n\n### L1 human\n\nThe test checks a property that every prime has, for a few random bases.\nA composite number fails for most bases, so a few rounds expose it with high probability.\nThe powers are fast to compute by [[modular exponentiation|04DR5PM25N]].\n",
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