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  "body": "\n## Summary\n\nThe sieve of Eratosthenes is an algorithm that finds all the [[prime numbers|0SN0MAEKYX]] up to a given bound $N$.\nIt writes the integers from $2$ to $N$. It takes the smallest number that is not crossed out, which is a prime $p$, and crosses out the multiples $p^2, p^2 + p, p^2 + 2p, \\dots$ up to $N$.\nIt repeats this step while $p^2 \\le N$. The numbers that are not crossed out at the end are the primes up to $N$.\n\n## Remarks\n\nIt is enough to start at $p^2$, because a smaller multiple of $p$ has a smaller prime factor, and an earlier step crossed it out.\nThe algorithm needs about $N \\log \\log N$ operations.\nThe idea of the sieve is the basis of sieve methods in analytic number theory.\n\n## Explanations\n\n### L1 human\n\nFor $N = 30$, cross out the multiples of $2$ from $4$, then the multiples of $3$ from $9$, then the multiples of $5$ from $25$.\nThen $7^2 = 49 \u003e 30$, so you stop.\nThe numbers left are $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$.\n",
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