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# Near Catalan's Equation

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.

Note by Worranat Pakornrat
1 month, 3 weeks ago

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no there isn't it is still a conjecture that there are finite many solutions to the equation $a^x - b^y = n$ where $n>1, n \in \mathbb{N}$

- 1 month, 3 weeks ago