20040627, 21:34  #1 
"Juan Tutors"
Mar 2004
569 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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