6
$\begingroup$

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?

$\endgroup$
12
$\begingroup$

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

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.