Catalan's Conjecture

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The only solution to the Diophantine equation:

$x^a - y^b = 1$

for $a, b > 1$ and $x, y > 0$, is:

$x = 3, a = 2, y = 2, b = 3$


Also known as

Catalan's Conjecture is now also known as Mihăilescu's Theorem, for Preda V. Mihăilescu who proved it true.

Also see

Source of Name

This entry was named for Eugène Charles Catalan.

Historical Note

Catalan's Conjecture was first put forward by Eugène Charles Catalan in $1844$.

The case where $x$ and $y$ are $2$ and $3$ was proved in $1344$ by Levi ben Gershon.

A proof was finally published in $2004$ by Preda V. Mihăilescu.