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If $a,b,c\in \mathbb Z$ are known and $a>b>1,(a,b)=1$, how many integer solutions are possible to the equation $$a^x-b^y=c~?\tag1$$ Can $(1)$ has more than $4$ integer solutions ?

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up vote 4 down vote accepted

There are at most two solutions in positive integers $x$ and $y$, due to some guy (Canadian J. Math 2001).``On some exponential equations of S. S. Pillai)''.

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Thanks, I remember it has at most two solutions, but not surely. –  Next Jul 24 '13 at 3:50

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