(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 $n^2 < p < (n + 1)^2$ 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 $n^2 < p < (n + 1)^2$ or a semiprime $n^2 < pq < (n + 1)^2$ (where $q$ is a prime unequal to $p$ . Thanks to computers, brute force searches have shown that the conjecture holds true as high as $n = 10^5$