20040627, 21:34  #1 
Mar 2004
3×163 Posts 
Question on prime powers
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. 
20040704, 12:53  #2  
Mar 2004
29 Posts 
Quote:
Catalan's theorem states that there is only one solution (3²2³) 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 

20040707, 07:07  #3 
Feb 2003
2^{5} Posts 
try this page: http://www.primepuzzles.net/conjectures/conj_031.htm

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