No proof or counterexample has ever been found, despite the claim being checked by computer for enormous ranges of n. Results exist for closely related but weaker statements, such as a prime between consecutive cubes for large enough n, but nobody has found a way to close the gap for consecutive squares, which is exactly the gap that makes it one of Landau's four unapproachable problems named in 1912.
What would resolve this A general proof covering every positive integer n, or a single confirmed counterexample: one gap between consecutive squares, however large, that a careful search shows contains no prime at all.
OpenAnalytic number theoryLegendre's Conjecture (Wikipedia)