Legendre’s conjecture

(Adrien-Marie Legendre) There is always a prime numberMathworldPlanetmath between a square number and the next. To put it algebraically, given an integer n>0, there is always a prime p such that n2<p<(n+1)2. Put yet another way, (π((n+1)2)-π(n2))>0, where π(x) is the prime counting function.

This conjecture was considered unprovable when it was listed in Landau’s problems in 1912. Almost a hundred years later, the conjecture remains unproven even as similar conjectures (such as Bertrand’s postulate) have been proven.

But progress has been made. Chen Jingrun proved a slightly weaker version of the conjecture: there is either a prime n2<p<(n+1)2 or a semiprime n2<pq<(n+1)2 (where q is a prime unequal to p). Thanks to computers, brute force searches have shown that the conjecture holds true as high as n=105.

Title Legendre’s conjecture
Canonical name LegendresConjecture
Date of creation 2013-03-22 16:38:17
Last modified on 2013-03-22 16:38:17
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 4
Author PrimeFan (13766)
Entry type Conjecture
Classification msc 11A41
Related topic BrocardsConjecture