Pillai's conjecture concerns a general difference of perfect powers (sequence A001597 in OEIS): it is an open problem initially proposed by S. S. Pillai, who conjectured that the gaps in the sequence of perfect powers tend to infinity. This is equivalent to saying that each positive integer occurs only finitely many times as a difference of perfect powers: more generally, in 1931 Pillai conjectured that for fixed positive integers A, B, C the equation has only finitely many solutions (x,y,m,n) with (m,n) ≠ (2,2). Pillai proved that the difference for any λ less than 1.
The general conjecture would follow from the ABC conjecture.
Paul Erdős conjectured that there is some positive constant c such that if d is the difference of a perfect power n, then d>nc for sufficiently large n.
Read more about this topic: Catalan's Conjecture
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