# Legendre’s conjecture

(Adrien-Marie Legendre) There is always a prime number between a square number and the next. To put it algebraically, given an integer $n>0$, there is always a prime $p$ such that $n^{2}. Put yet another way, $(\pi((n+1)^{2})-\pi(n^{2}))>0$, where $\pi(x)$ 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 $n^{2} or a semiprime $n^{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 $n=10^{5}$.

Title Legendre’s conjecture LegendresConjecture 2013-03-22 16:38:17 2013-03-22 16:38:17 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Conjecture msc 11A41 BrocardsConjecture