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A few days ago a question was asked about is there always at least one prime in closed intervals of the form $[2^n,2^n+n^2]$ (endpoints included) and, current state-of-the-art/science, is that we still do not know..

I was thinking a little about more general problem, that is, about primes in intervals of the form $[a^n,a^n+n^2]$ where $a\geq2$ is an integer.

I am supposing, with very little of knowledge about this field, that also this more general problem is open for every $a\geq2$.

But also, it somehow seems irrelevant to me what $a$ we will choose, that is, that "almost always" there will be a prime in such intervals, where, of course, "almost always" means that density of such intervals that contain no primes equals zero. (There is no need at the present moment to go into different possible (equivalent or non-equivalent) definitions of densities)

But even if we do not know is that true for every $a \geq 2$, what can be told about occurences of primes "on average" for different $a$? Should the density be equal for every $a$ we choose(even if it is not equal to $1$, but it seems unavoidable that it is)? Is it equal for every $a$?

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  • $\begingroup$ Cramer's conjecture asserts that the gap between $p_n$ and $p_{n+1}$, where $p_n$ is the $n$-th prime, does not exceed $O((\log p_n)^2)$. It is not clear what the precise term in the big-$O$ term should be, and that essentially determines what value of $a$ one should expect there to always be a prime in an interval of that shape. On average prime gaps are only $O(\log p_n)$ in size, so for 'most' intervals of that shape there should be at least one prime in them. $\endgroup$ – Stanley Yao Xiao Dec 26 '17 at 18:21
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There is no prime between $11^3$ and $11^3+9$. I found it by inspecting this table for gaps starting near powers. A near miss is $5^6$. I wonder if there are infinitely many for some $a$ — it wouldn't violate the effective form of Cramér's conjecture, and there are already examples of gaps near but not quite exceeding $\log_e(p)^2$.

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  • $\begingroup$ That does not violate this in "almost always" sense. $\endgroup$ – user114642 Dec 26 '17 at 18:41
  • $\begingroup$ Yeah reading your question more closely, I'm simply rephrasing it. I'll leave this answer here since at least it has an example. $\endgroup$ – Dan Brumleve Dec 26 '17 at 18:46
  • $\begingroup$ However, if there is a prime in the interval infinitely often for some $a$, that would provide an improvement to my finding primes infinitely-often conjecture which I asked about yesterday. But I don't have any real evidence that my conjecture is open yet, other than failing to solve it or reduce it to another open problem myself. $\endgroup$ – Dan Brumleve Dec 26 '17 at 18:49
  • $\begingroup$ It would be interesting to at least know is there a finite or infinite number of counterexamples for all $a\geq2$ and all $n$. You found one. $\endgroup$ – user114642 Dec 26 '17 at 19:03
  • $\begingroup$ I'm asking the weaker question now, is there an $a$ with an infinite number of examples? (If so we can find primes infinitely often in polynomial time answering my other question.) $\endgroup$ – Dan Brumleve Dec 26 '17 at 19:05

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