Pythagorean Triple

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 (PPT) is one in which a, b and c are pairwise coprime. A right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle.

The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a2 + b2 = c2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. 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 do not have an integer common multiple because √2 is irrational.

Read more about Pythagorean Triple:  Examples, Generating A Triple, Elementary Properties of Primitive Pythagorean Triples, Some Relationships, A Special Case: The Platonic Sequence, Geometry of Euclid's Formula, Spinors and The Modular Group, Parent/child Relationships, Relation To Gaussian Integers, Distribution of Triples

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