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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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    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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For your first question, if we suppose that $a, b \geq 2$ and $k$ are all fixed, then there are at most two solutions in positive exponents $m$ and $n$. This follows from lower bounds for linear forms in logarithms (and probably other approaches). As for Pillai's conjecture, it's wide open still (modulo developments on the ABC conjecture). It is still unknown, by way of example, whether there are only finitely many perfect powers differing by $2$.

Standard conjectures would imply that your function $f(d)$ is zero for "most" $d$ (if we agree to avoid $m=n=2$).