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I am looking for consecutive entries $(a_k,a_{k+1})$ in the sequence of prime powers $$(a_n)=(2,3,2^2,5,7,2^3,3^2,11,13,2^4,17,19,23,5^2,3^3,29,31,2^5\cdots)$$ such that neither $a_k$ nor $a_{k+1}$ are primes. Examples are $(2^3,3^2), (5^2,3^3), (3^7,13^3)$. I realize this may be a very difficult problem, but I wonder how many have been found and what is known.

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    $\begingroup$ oeis.org/A025474 $\endgroup$
    – healynr
    Oct 18 at 16:37
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    $\begingroup$ A068435 (Consecutive prime powers without a prime between them). $\endgroup$
    – mathlove
    Oct 18 at 16:44
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    $\begingroup$ Following the links provided in the comments, I have learned that only five such pairs are known to exist: $(2^3,3^2),(5^2,3^3),(11^2,5^3),(3^7,13^3),(181^2,2^{15})$. $\endgroup$
    – Valerio
    Oct 18 at 19:37
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    $\begingroup$ Pillai's conjecture predicts large differences between perfect powers (apart from finite many exceptions) and this makes it likely that a prime is between such consecutive prime powers with a few expections. I do not know how far this has been checked, I guess upto at least $10^{12}$. I agree that the problem is very difficult and proofs are not to be expected. $\endgroup$
    – Peter
    Oct 19 at 8:19
  • $\begingroup$ Also the ratio must be less than 2 ( or 1.05 once big enough) $\endgroup$ Oct 20 at 14:07

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