# What is the longest run of consecutive natural numbers that are prime powers?

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.

• 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: $2$, $3$, $4$, $5$. – Michael Burr Mar 1 '17 at 2:29
• I think the question might be more interesting if you look at "consecutive odd numbers" – lulu Mar 1 '17 at 2:33
• @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. – btilly Mar 1 '17 at 2:35
• @btilly Ah, quite right. – lulu Mar 1 '17 at 2:36
• 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 $8, 9.$ – B. Goddard Mar 1 '17 at 11:57