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单词 BrocardsConjecture
释义

Brocard’s conjecture


(Henri Brocard) With the exception of 4 and 9, there are always at least four prime numbersMathworldPlanetmath between the square of a prime and the square of the next prime. To put it algebraically, given the nth prime pn (with n>1), the inequality (π(pn+12)-π(pn2))>3 is always true, where π(x) is the prime counting function.

For example, between 22 and 32 there are only two primes: 5 and 7. But between 32 and 52 there are five primes: a prime quadrupletMathworldPlanetmath (11, 13, 17, 19) and 23.

This conjecture remains unproven as of 2007. Thanks to computers, brute force searches have shown that the conjecture holds true as high as n=104.

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更新时间:2025/5/4 16:27:23