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  "body": "\n## Summary\n\nThe Fermat primality test is a probabilistic test of whether an integer $n \u003e 1$ is prime.\nIt chooses a base $a$ with $1 \u003c a \u003c n - 1$ and computes $a^{n-1} \\bmod n$ by [[modular exponentiation|04DR5PM25N]].\nIf the result is not $1$, then $n$ is composite, by [[Fermat's little theorem|0M25PRN3WQ]]. If the result is $1$, then $n$ is called a probable prime to the base $a$.\n\n## Remarks\n\nA probable prime can be composite. Such a number is a [[Fermat pseudoprime|02M2K4QH9P]].\nThe [[Carmichael numbers|0TA7WT68RK]] pass the test for each base coprime to them. So repeated tests with many bases cannot detect all composite numbers.\nThe [[Miller–Rabin primality test|0SZCSYHW60]] is a stronger test of the same kind, without this weakness.\n\n## Explanations\n\n### L1 human\n\nTo test $n = 15$ with $a = 2$, compute $2^{14} = 16384 \\equiv 4 \\pmod{15}$.\nThe result is not $1$, so $15$ is surely composite. The test does not find a factor; it only proves that one exists.\n",
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