The Pythagorean theorem: a² + b² = c²

A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime. Image File history File links Illustration of the Pythagorean theorem Wapcaplet scaled down the image, and added color and labels. ... Image File history File links Illustration of the Pythagorean theorem Wapcaplet scaled down the image, and added color and labels. ... Jump to: navigation, search The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... Coprime - Wikipedia /**/ @import /skins-1. ...

The name is derived from the Pythagorean theorem, of which every Pythagorean triple is a solution. The converse is not true. For instance, the triangle with sides a = b = 1 and c = √2 is right, but (1, 1, √2) is not a Pythagorean triple because √2 is not an integer. Moreover, 1 and √2 don't have an integer common multiple because √2 is irrational. The Pythagorean theorem: The sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse. ... Jump to: navigation, search A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments. ... Jump to: navigation, search In mathematics, an irrational number is any real number that is not a rational number, i. ...

There are 16 primitive Pythagorean triples with c ≤ 100:

Generating Pythagorean triples

An effective way to generate Pythagorean triples is based on the observation that if m and n are two positive integers with m > n, then

a = m² − n²,

b = 2mn,

c = m² + n²

is a Pythagorean triple. It is primitive if and only if m and n are coprime and one of them is even (if both n and m are odd, then a, b, and c will be even, and so the Pythagorean triple will not be primitive). Not every Pythagorean triple can be generated in this way, but every primitive triple (possibly after exchanging a and b) arises in this fashion from a unique pair of coprime numbers m > n. This shows that there are infinitely many primitive Pythagorean triples. Coprime - Wikipedia /**/ @import /skins-1. ...

See also

• Heronian triangle
• Platonic sequence

A Heronian triangle is a triangle whose side lengths and area are all rational numbers. ... // History Odd Variation Even Variation The Platonic sequence is one of the first rational arithmetic triangle theorems of the ancient world. ...

External links

• http://mathworld.wolfram.com/PythagoreanTriple.html has an extensive discussion of Pythagorean triples.
• http://www.math.clemson.edu/~rsimms/neat/math/pyth/ provides a Javascript calculator for the (m² − n², 2mn, m² + n²) formula, and shows how to derive the formula.
• http://www.faust.fr.bw.schule.de/mhb/pythagen.htm a JavaScript calculator which illustrates the 3-fold tree structure of the set of all primitive Pythagorean triples.
• Pythagorean Triples Where the formula comes from: interactive Java illustration.
• The Trinary Tree(s) underlying Primitive Pythagorean Triples by H. Andres Lönnemo
• Fermat's Last Theorem Blog Covers topics in the history of Fermat's Last Theorem from Pythagorean triples to Wiles' proof.