Fermat's right triangle theorem

No right triangle with integer side lengths has an area that is a perfect square. The result is also known as Fermat's theorem on right triangles.

Infinite descent: A proof method that derives an impossible infinite sequence of strictly decreasing positive integers. Fermat's proof rules out square-area triangles by repeatedly constructing a smaller one.
Heronian triangle: A triangle with integer side lengths and integer area. The theorem excludes square areas, a special kind of integer area, among right Heronian triangles.
Pierre de Fermat: A French mathematician whose work helped establish number theory as a major field. He stated the theorem and is credited with its infinite-descent proof.
Birch and Swinnerton-Dyer conjecture: A conjecture relating an elliptic curve's rational points to the behavior of its L-function at one. The congruent-number curves connected to this theorem fall within the conjecture's scope.
Pythagorean triple: Three positive integers that satisfy the equation a² + b² = c². The theorem concerns the areas of right triangles whose side lengths form such a triple.
Integer square: An integer of the form n² for some integer n. The prohibited area is specifically a perfect square, not merely an integer.
Fermat's Last Theorem: The assertion that xⁿ + yⁿ = zⁿ has no positive integer solutions for n greater than two. Both results reflect Fermat's use of descent to prohibit integer solutions.
Tunnell's theorem: A theorem giving a computable criterion for whether a squarefree integer is a congruent number, conditional on Birch and Swinnerton-Dyer. For square numbers, its criterion reflects the absence of integer-sided square-area right triangles.
Primitive Pythagorean triple: A Pythagorean triple whose three entries have no common divisor greater than one. Any square-area counterexample can be reduced to a primitive triple before descent.
Diophantine equation: An equation for which integer or rational solutions are sought. The triangle's side lengths and square area impose simultaneous integer equations.

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