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Old 2004-06-27, 21:34   #1
JuanTutors
 
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Mar 2004

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Default 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.
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Old 2004-07-04, 12:53   #2
juergen
 
Mar 2004

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Quote:
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
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Old 2004-07-07, 07:07   #3
1260
 
Feb 2003

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