Define two prime powers $p^m<q^n$, ($m,n\in\mathbb N$) to be consecutive if it doesn't exist a prime power $r^k$ such that $p^m\!<r^k\!<q^n$. It seems that every even number is the difference between two consecutive prime powers, I haven't ran any large tests yet, but what about the odd numbers?

Are there two consecutive prime powers $p^m<q^n$ and $i\in\mathbb N^+$ such that $q^n-p^m=11^i$?

I have tested all consecutive prime powers less than $10,000,000$ without finding any solution to $q^n-p^m=11^i$.

(A pattern seems to suggest that no odd numbers $N\ge 31$ have a solution to $q^n-p^m=N$, consecutive prime powers. But this must be wrong since prime powers has similar asymptotic distribution as the primes).

  • $\begingroup$ This is a good question. But "power", not "potency", is the usual word in English. I've edited accordingly. $\endgroup$ – Michael Lugo Mar 3 '17 at 18:13
  • $\begingroup$ @MichaelLugo, thanks! $\endgroup$ – Lehs Mar 3 '17 at 18:13
  • $\begingroup$ So one of the primes has to be even, and there are few of those. $\endgroup$ – hardmath Mar 3 '17 at 18:15
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    $\begingroup$ For powers of $11$ greater than the first we get into the realm of the abc-conjecture. $\endgroup$ – hardmath Mar 3 '17 at 18:21
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    $\begingroup$ 'No odd numbers $N\geq 31$ have a solution to $q^n-p^m=N$' is almost certainly false heuristically; near $2^k$ the probability that the next $N$ numbers are composite is approximately $(1-\frac{\ln 2}k)^{N/2}$ and for all $N$ this is bounded away from zero for large $k$, so we should in fact expect infinitely many powers of $2$ whose nearest prime is $\geq N$ away from them. $\endgroup$ – Steven Stadnicki Mar 3 '17 at 22:47

See the OEIS: https://oeis.org/A013603 gives the difference between $2^n$ and the largest prime less than $2^n$. We can read off that:

  • $2^n - 11$ is prime and there are no primes between it and $2^{n}$ for $n = 42, 78, 114, 190, 322, 546, 3894$.
  • $2^n - 11^2$ is prime and there are no primes between it and $2^n$ for $n = 219, 303, 1443, 2333, 2589, 3315, 3693$.
  • $2^n - 11^3$ is prime and there are no primes between it and $2^n$ for $n = 1390, 1670$.

These are the only examples where $n < 5000$ and $q = 2$. The "mirror image" sequence of the gap between $2^m$ and the smallest prime greater than $2^m$, which doesn't exist in OEIS, would be needed for the $p = 2$ case.

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    $\begingroup$ For example, $139-2^7=11$, but $137$ was already prime, and thus these prime powers are not consecutive. $\endgroup$ – hardmath Mar 3 '17 at 18:36
  • $\begingroup$ See OEIS A013597 for something of a "mirror image" sequence of a(n) = nextprime(2^n) - 2^n. $\endgroup$ – hardmath Mar 21 '17 at 21:29

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