TITLE: Near-Legendre Conjecture
QUESTION [7 upvotes]: 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?

REPLY [13 votes]: Baker, Harman and Pintz showed in 2001 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