Ingham has shown that there is a prime between $n^{3}$ and $(n+1)^{3}$ for large enough $n.$

Legendre's conjecture about the existence of primes between consecutive perfect squares is of course open.

What, if anything, is known about the existence of primes in the intervals $$ [n^{2+\epsilon},(n+1)^{2+\epsilon}], $$ for $n$ large enough?