By the result of Baker, Harman, Pintz (http://www.cs.umd.edu/~gasarch/BLOGPAPERS/BakerHarmanPintz.pdf), for any sufficiently large $x$ the interval $[x-x^{21/40},x]$ contains a prime number. This result implies the asymptotic $p(x)=x+O(x^{21/40})$ where the function $p(x)$ assigns to each real number $x$ the smallest prime number $p\ge x$.

**Question.** For which smallest possible constant $\theta$ is it known that $[x-x^\theta,x]$ contains a power of a prime number? Can this $\theta$ be smaller or equal than $\frac12$?

This problem admits also the following reformulation. For any real number $x$ let $q(x)$ be the smallest prime power greater or equal than $x$.

**Problem.** Is the asymptotic growth of the function $q(x)$ better than that of $p(x)$. For example, is $q(x)=x+O(\sqrt{x})$? Is this equality true under the Riemannian Hypothesis?

`6434856, 6436343, 6436369`

;`27027009001, 27027031201, 27027081632`

;`34359738368, 34359812496, 34359822251`

;`42618264157, 42618299364, 42618442977`

;`312079600999, 312079650687, 312079766881`

;`328080365089, 328080401001, 328080696273`

;`11305786504384, 11305787424768, 11305787558464`

. But as you said it's unclear whether there are infinitely many of them. $\endgroup$ – WhatsUp Feb 14 '17 at 15:45