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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?

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    $\begingroup$ Considering that the density of prime powers is so thin even within the primes, it seems as though results any better than the one for primes would be hard. In particular there are only $o(\sqrt{x})$ prime powers $\lt x$ that aren't primes themselves so it would be very surprising if there were enough of them to improve the constant as far as you want it to go. $\endgroup$
    – Steven Stadnicki
    Commented Feb 14, 2017 at 6:08
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    $\begingroup$ I think it is unknown if there are three integer powers occurring within a square root of n for infinitely many n. Only 2187,2197,2209 comes to mind. (Oh, and 121,125,128.) Gerhard "Integer Powers Are Somewhat Scarce" Paseman, 2017.02.13. $\endgroup$
    – Gerhard Paseman
    Commented Feb 14, 2017 at 6:09
  • $\begingroup$ I have just discovered that the same question has been already discussed in mathoverflow.net/questions/111029/prime-power-gaps?rq=1 $\endgroup$
    – Taras Banakh
    Commented Feb 14, 2017 at 15:05
  • $\begingroup$ @GerhardPaseman Here are some more: 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
    Commented Feb 14, 2017 at 15:45
  • $\begingroup$ @WhatsUp Could you explain what those numbers (like x=6434856) mean? That the interval $[x-\sqrt{x},x]$ or $[x,x+\sqrt{x}]$ contains no prime powers? Or something else? $\endgroup$
    – Taras Banakh
    Commented Feb 14, 2017 at 15:50

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