So, suppose that we have a natural number,n, that is a prime power. Is there a largest k such that run like this: $n=p^{a_i}_i$,$n+1=p^{a_i}_i$,$n+2=p^{a_i}_i$...$n+k=p^{a_i}_i$, only goes so far? I was playing around with Mathematica and it seems that for any distance there might be infinitely many prime powers that are within that distance close to each other. I think that my specified distance of 1 can't have a run of length more than two, but am not sure how to prove this. Any help would be appreciated.

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    $\begingroup$ If $n$ is even, then $n+2$ is also even. It is very rare for both $n$ and $n+2$ to be powers of $2$. The answer is $4$ from the run: $2$, $3$, $4$, $5$. $\endgroup$ – Michael Burr Mar 1 '17 at 2:29
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    $\begingroup$ I think the question might be more interesting if you look at "consecutive odd numbers" $\endgroup$ – lulu Mar 1 '17 at 2:33
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    $\begingroup$ @lulu You would just apply the same logic with 3 to put an upper bound on it. And the answer is 3, 5, 7, 9, 11, 13. $\endgroup$ – btilly Mar 1 '17 at 2:35
  • $\begingroup$ @btilly Ah, quite right. $\endgroup$ – lulu Mar 1 '17 at 2:36
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    $\begingroup$ I think the term "prime power" usually means the exponent is greater than 1. So I think the only answer here is $k=2$ for $8, 9.$ $\endgroup$ – B. Goddard Mar 1 '17 at 11:57

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