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Given positive integers k,a,b, is there a finite or infinite number of solutions in positive integers $m,n>1$, to $a^m+k=b^n$?


Pillai's conjecture states that each positive integer occurs only finitely many times as a difference of perfect powers (Only k given, a,m,n,b are variables) . It is an open problem.

What are known lower bounds on f(d) defined as how many times d, for d=1,2,3... occurs as a difference of perfect powers?

Catalan's conjecture is the theorem that f(1)=1

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    Do you mean $a^{m+1}$ or $a^{m}+1$?2012-02-06
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    http://en.wikipedia.org/wiki/Catalan%27s_conjecture2012-02-06
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    In spite of the name, Catalan's conjecture is actually a theorem. Your second question differs from Pillai's conjecture by fixing $a$ and $b$, which is likely to make the answer always "finite". You should be clearer about which question exactly you are asking.2012-02-06
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    Please avoid double use of letters for different things. Why not use "d" for the difference instead of "a" (which already means one of the two bases in the problemdescription)?2012-02-06
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    Is it required that m and n are different? If it is permitted that m = n = 2, a non-zero lower bound can be found for most k as follows. Find the prime factors of k, then use these to find all pairs p, q such that pq = k and p > q. If k is even, consider only those where p, q are both even. Then use each such pair to find a series of q consecutive odd integers with mean p and therefore summing to k. The sum of any such series must be the difference between two squares, ie betwen ((p+q)/2)^2 and ((p-q)/2)^2.2012-02-10
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    I think the abc-conjecture (about which there is much literature on the web and elsewhere) implies there are only finitely many solutions. But of course the abc-conjecture is a conjecture, not a theorem.2012-04-23
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    This probably follows from Siegel's theorem directly.2013-01-29
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    A related question here http://math.stackexchange.com/questions/1452855/2n2-lfloor-mb-rfloor-k-has-only-finitely-many-integer-solutions2015-09-27

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