From Catalan's theorem, we know that \(3^2 - 2^3 = 1\) is the only solution to the following equation, for some integers \(a,b \geq 1\) and \(x,y \geq 2\):

\[a^x - b^y = 1\]

Now is there any theorem confirming for \(3^3 - 5^2 = 2\) as the the only solution to the equation:

\[a^x - b^y = 2\]

as well?

Any reference to any paper would be appreciated. Thank you.

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TopNewestno there isn't it is still a conjecture that there are finite many solutions to the equation \[a^x - b^y = n\] where \[n>1, n \in \mathbb{N} \]

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