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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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)''.