# Consecutive prime powers that are not prime

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.

• oeis.org/A025474 Oct 18 at 16:37
• A068435 (Consecutive prime powers without a prime between them). Oct 18 at 16:44
• Following the links provided in the comments, I have learned that only five such pairs are known to exist: $(2^3,3^2),(5^2,3^3),(11^2,5^3),(3^7,13^3),(181^2,2^{15})$. Oct 18 at 19:37
• 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. Oct 19 at 8:19
• Also the ratio must be less than 2 ( or 1.05 once big enough) Oct 20 at 14:07