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For relatively prime integers $a$ and $b$ both greater than one and oddinteger $c$, there are at most two solutions in positive integers $(x,y,z)$ tothe equation $a^x + b^y = c^z$. There are an infinite number of $(a,b,c)$giving exactly two solutions.

Number of solutions to $a^x + b^y = c^z$, A Shorter Version