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  "body": "\n## Summary\n\nThe Brun sieve is a combinatorial sieve method of Viggo Brun.\nIt counts the integers of a set that have no prime factor below a bound $z$.\nIt uses a truncated inclusion–exclusion, which gives upper and lower bounds with a controllable error.\nIt was the first sieve to give nontrivial results on problems such as twin primes.\nThere the [[sieve of Eratosthenes|0XM8ZA9RNM]] with full inclusion–exclusion produces too large an error.\nWith it Brun proved that the sum of the reciprocals of the twin primes converges.\n\n## Explanations\n\n### L1 human\n\nTo count numbers with no small prime factor, inclusion–exclusion subtracts the multiples of each prime, adds back the multiples of each product of two primes, and so on.\nWith many primes this formula has a huge number of terms, and the small rounding errors add up to something larger than the answer.\nBrun's idea is to stop the formula early.\nStopping after an even number of steps gives an upper bound, stopping after an odd number gives a lower bound.\nFor example, the numbers up to $10$ with no factor $2$ or $3$ are $1, 5, 7$; stopping after the first step gives the lower bound $10 - 5 - 3 = 2 \\le 3$.\n",
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