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?



Near-Legendre Conjecture
Baker, Harman and Pintz [showed in 2001](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.360.3671&rep=rep1&type=pdf) that for $\theta=0.525$, the interval $(x,x+x^\theta)$ contains at least one prime provided $x$ is sufficiently large. This is equivalent to the interval $[n^{2+\epsilon},(n+1)^{2+\epsilon}]$ to contain a prime for $\epsilon=\frac{2\theta-1}{1-\theta}\approx 0.1053$ and $n$ sufficiently large


