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