Fermat's right triangle theorem

Fermat's right triangle theorem is a non-existence proof in number theory, published in 1670 among the works of Pierre de Fermat, soon after his death.

[1] It has many equivalent formulations, one of which was stated (but not proved) in 1225 by Fibonacci.

In its geometric forms, it states: More abstractly, as a result about Diophantine equations (integer or rational-number solutions to polynomial equations), it is equivalent to the statements that: An immediate consequence of the last of these formulations is that Fermat's Last Theorem is true in the special case that its exponent is 4.

In 1225, Emperor Frederick II challenged the mathematician Fibonacci to take part in a mathematical contest against several other mathematicians, with three problems set by his court philosopher John of Palermo.

In The Book of Squares, published later the same year by Fibonacci, he solved the more general problem of finding triples of square numbers that are equally spaced from each other, forming an arithmetic progression.

Fibonacci called the gap between these numbers a congruum.

[2] One way of describing Fibonacci's solution is that the numbers to be squared are the difference of legs, hypotenuse, and sum of legs of a Pythagorean triangle, and that the congruum is four times the area of the same triangle.

[3] Fibonacci observed that it is impossible for a congruum to be a square number itself, but did not present a satisfactory proof of this fact.

could form an arithmetic progression whose congruum was also a square

That is, by the Pythagorean theorem, they would form two integer-sided right triangles in which the pair

But if (as Fibonacci asserted) no square congruum can exist, then there can be no two integer right triangles that share two sides in this way.

[5] Because the congrua are exactly the numbers that are four times the area of a Pythagorean triangle, and multiplication by four does not change whether a number is square, the existence of a square congruum is equivalent to the existence of a Pythagorean triangle with a square area.

It is this variant of the problem that Fermat's proof concerns: he shows that there is no such triangle.

In considering this problem, Fermat was inspired not by Fibonacci but by an edition of Arithmetica by Diophantus, published in a translation into French in 1621 by Claude Gaspar Bachet de Méziriac.

Conversely, any three positive integers obeying the equation

Therefore, Fermat's proof that no Pythagorean triangle has a square area implies the truth of the exponent-

[9] Another more geometric way of stating this formulation is that it is impossible for a square (the geometric shape) and a right triangle to have both equal areas and all sides commensurate with each other.

[10] Yet another equivalent form of Fermat's theorem involves the elliptic curve consisting of the points whose Cartesian coordinates

The points (−1,0), (0,0), and (1,0), provide obvious solutions to this equation.

Fermat's theorem is equivalent to the statement that these are the only points on the curve for which both

More generally, the right triangles with rational sides and area

correspond one-for-one with the rational points with positive

[11] During his lifetime, Fermat challenged several other mathematicians to prove the non-existence of a Pythagorean triangle with square area, but did not publish the proof himself.

Fermat's son Clement-Samuel published an edition of this book, including Fermat's marginal notes with the proof of the right triangle theorem, in 1670.

It shows that, from any example of a Pythagorean triangle with square area, one can derive a smaller example.

Since Pythagorean triangles have positive integer areas, and there does not exist an infinite descending sequence of positive integers, there also cannot exist a Pythagorean triangle with square area.

are the integer sides of a right triangle with square area.

, by which the problem is transformed into finding relatively prime integers

are the legs of another primitive Pythagorean triangle whose area is

Thus, any Pythagorean triangle with square area leads to a smaller Pythagorean triangle with square area, completing the proof.