Fermat's right triangle theorem

Statement

There is no right triangle with integer side lengths whose area is the square of an integer. Equivalently, the equation $x^4 - y^4 = z^2$ has no solution in positive integers $x$, $y$ and $z$.

Remarks

In terms of a Pythagorean triple $(a, b, c)$, the theorem says that $ab/2$ is never a perfect square. A consequence is Fermat's Last Theorem for n = 4, since $a^4 + b^4 = c^4$ gives $c^4 - b^4 = (a^2)^2$. The proof is by infinite descent.

History

The proof was published in 1670 among the works of Pierre de Fermat, after his death. It is the only complete proof by Fermat that survives.

Explanations

L1 human

The triangle with sides $3$, $4$, $5$ has area $6$, and the one with sides $5$, $12$, $13$ has area $30$. Fermat proved that such an area can never be a perfect square.