Catalan"s Conjecture

Theorem

The only solution to the Diophantine equation:

$x^a - y^b = 1$

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

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

Proof

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Also known as

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

Also see

• Consecutive Integers which are Powers of 2 or 3: the special case where $x$ and $y$ are $2$ and $3$
• 1 plus Power of 2 is not Perfect Power except 9: the special case of $y = 2$.
• 1 plus Perfect Power is not Power of 2: the special case of $x = 2$.
• 1 plus Square is not Perfect Power: the special case of $b = 2$.
• 1 plus Perfect Power is not Prime Power except for 9: the special case where $x$ is prime.

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.

Sources

• 1944: Eugène Catalan: Note extraite d"une lettre adressée à l"éditeur (J. reine angew. Math. Vol. 27: p. 192)
• 2004: Preda Mihăilescu: Primary Cyclotomic Units and a Proof of Catalan"s Conjecture (J. reine angew. Math. Vol. 572: pp. 167 – 195)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Catalan"s conjecture