# Near-Legendre Conjecture

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?

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