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  "body": "\n## Statement\n\nLet $p$ be an odd prime, and let $M_p = 2^p - 1$ be the [[Mersenne number|0GJBJ19NXR]].\nDefine the sequence $(s_i)_{i \\ge 0}$ by $s_0 = 4$ and $s_i = s_{i-1}^2 - 2$ for $i \\ge 1$.\nThen $M_p$ is prime if and only if $s_{p-2} \\equiv 0 \\pmod{M_p}$.\n\n## Remarks\n\nFor example, take $p = 5$ and $M_5 = 31$. Modulo $31$, the sequence is $4, 14, 8, 0$, so $s_3 \\equiv 0$ and $31$ is prime.\nIn practice, one reduces each term modulo $M_p$. Then the test needs $p - 2$ squarings of numbers with at most $p$ bits.\n\n## History\n\nÉdouard Lucas developed the test in 1878, and Derrick Henry Lehmer proved it in 1930.\n\n## Explanations\n\n### L1 human\n\nThe test checks a Mersenne number for primality with a simple loop: square, subtract $2$, and reduce modulo $M_p$.\nAfter $p - 2$ steps, the result is $0$ exactly when $M_p$ is prime.\n",
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