Question on prime powers

[JuanTutors]

I'm wondering if it can be shown, or even if it's known yet, whether there exist primes p,q such that (p^m)-(q^n)<=A for an arbitrary positive integer A.

I am wondering because, in the special case where p and q are 2 and 3 (or vice versa) it might be easy to prove primality or compositeness of integers N in the range (q^n)<N<(p^m) through arguments about what forms the factors of N must have.

[juergen]

Originally Posted by dominicanpapi82

I'm wondering if it can be shown, or even if it's known yet, whether there exist primes p,q such that (p^m)-(q^n)<=A for an arbitrary positive integer A.

I am wondering because, in the special case where p and q are 2 and 3 (or vice versa) it might be easy to prove primality or compositeness of integers N in the range (q^n)<N<(p^m) through arguments about what forms the factors of N must have.

Sorry, I don't know the answer to your question, but maybe you would like to read about Catalan's theorem which is the special case A=1.

Catalan's theorem states that there is only one solution (3²-2³) for this equation. It has been prooven a few years ago after about 150 years of uncertainty :o)
So I guess the answer to your question is not trivial.

Jürgen

[1260]

try this page: http://www.primepuzzles.net/conjectures/conj_031.htm