Diophantine Equation y cubed equals x squared plus 2

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Theorem

$y^3 = x^2 + 2$

has only one integer solution:

$x = 5, y = 3$

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 $1659$.