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

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also known as

This result is also known as Mihăilescu's Theorem, for Preda V. Mihăilescu.

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 $1844$.

It was proven in $2002$ 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