Diophantine Equation y cubed equals x squared plus 2

Theorem

The indeterminate Diophantine equation:

$y^3 = x^2 + 2$

has only one integer solution:

$x = 5, y = 3$

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.

Historical Note

The Diophantine equation $y^3 = x^2 + 2$ was proved to have only the solution $x = 5, y = 3$ by Pierre de Fermat by use of the Method of Infinite Descent.

He submitted it, without proof, along with a number of others, to Pierre de Carcavi in a letter dated $14$ August $1659$.

Sources

• 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{IV}$: The Prince of Amateurs