---
tag: 086BE0J3FZ
kind: statement
role: theorem
name: Fermat's right triangle theorem
msc: [11D25]
uses: [01G1WY1GVX, 0EFYCFHYFG]
sources:
  - url: https://en.wikipedia.org/wiki/Fermat%27s_right_triangle_theorem
    locator: Formulation
same_as:
  wikidata: Q18640217
  wikipedia: 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|01G1WY1GVX]] $(a, b, c)$, the theorem says that $ab/2$ is never a perfect square.
A consequence is [[Fermat's Last Theorem for n = 4|0E741HSMJ1]], since $a^4 + b^4 = c^4$ gives $c^4 - b^4 = (a^2)^2$.
The proof is by [[infinite descent|0EFYCFHYFG]].

## 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.