Fermat's right triangle theorem

Builds on Pythagorean triple Proof by infinite descent

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.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Divisibility definition
Proof by infinite descent concept
Greatest common divisor definition
Pythagorean triple definition
Fermat's right triangle theorem statement

Where this leads

2 pages build on this one.

Tag	`086BE0J3FZ`
URI	`m2://086BE0J3FZ`
Kind	statement, theorem
Writing	W1 stated
MSC	11D25
Status	This copy of the site has no build data. Run `m2 status 086BE0J3FZ`.
Uses	Pythagorean triple, Proof by infinite descent
Same as	Wikidata Q18640217 Wikipedia Fermat's right triangle theorem

Fermat's right triangle theorem

Statement

Remarks

History

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

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