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.

Title Brocard’s conjecture
Canonical name BrocardsConjecture
Date of creation 2013-03-22 16:40:53
Last modified on 2013-03-22 16:40:53
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 8
Author PrimeFan (13766)
Entry type Conjecture
Classification msc 11A41
Synonym Brocard conjecture
Related topic LegendresConjecture