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I came up with an interesting mathematical conjecture: for every natural number $n$ there is only a finite number of integer powers $a^x$ and $b^y$ such that $b^y - a^x = n$.

I would like to find out what it is called and who first came up with it.

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    $\begingroup$ This is known as Pillai's conjecture, a generalisation of Catalan's conjecture (en.wikipedia.org/wiki/Catalan%27s_conjecture) $\endgroup$
    – Thomas Bloom
    Commented Oct 25, 2023 at 8:10
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    $\begingroup$ @ThomasBloom Catalan's conjecture is not covered by Pillai's conjecture, since the latter only says that there are finitely many solutions (and not just the obvious ones). $\endgroup$
    – YCor
    Commented Oct 25, 2023 at 8:14
  • $\begingroup$ Depends a bit on what you call an "integer power". $17=18^1-1^1=19^1-2^1=\dotsb=5^2-8^1=6^2-19^1=\dotsb$. $\endgroup$
    – Gerry Myerson
    Commented Oct 26, 2023 at 10:16

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