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The sieve of Eratosthenes
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The sieve in which Eratosthenes separates the prime numbers is infinitely time-consuming. But it does have the slow excitement of hunting an increasingly rare species.
He writes out an open-ended sequence of odd numbers, then crosses out every third number after 3, every fifth after 5, every seventh after 7 and so on. Any number which does not eventually get crossed out is by definition prime, however long it may take to reach it.
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