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  "body": "\n## Definition\n\nA Mersenne prime is a [[prime number|0SN0MAEKYX]] of the form $2^n - 1$ for a natural number $n$.\nThat is, a Mersenne prime is a [[Mersenne number|0GJBJ19NXR]] that is prime.\n\n## Remarks\n\nThe first Mersenne primes are $3$, $7$, $31$ and $127$, for $n = 2, 3, 5, 7$.\nIf $2^n - 1$ is prime, then [[$n$ is prime|07MMWAXMYD]]. The converse fails: $2^{11} - 1 = 2047 = 23 \\cdot 89$.\nThe [[Lucas–Lehmer test|061BTYA0RN]] decides whether $2^p - 1$ is prime.\nIt is not known whether there are infinitely many Mersenne primes.\n\n## Explanations\n\n### L1 human\n\nA Mersenne prime is a prime that is one less than a power of $2$, such as $31 = 2^5 - 1$.\nThe even [[perfect numbers|0SB7Q8SYSF]] come exactly from these primes, by the [[Euclid–Euler theorem|0WETJCKNRG]].\n",
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