page_title: Pythagorean triple
section_title: Examples
hierarchical_section_title: Pythagorean triple / Examples
caption_attribution_description: English: Scatter plot of the legs (a,b) of the first Pythagorean triples with a and b less than 6000.
context_page_description: A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². Such a triple is commonly written, and a well-known example is. If is a Pythagorean triple, then so is for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right 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 a right triangle, but 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.
  Pythagorean triples have been known since ancient times. The oldest known record comes from Plimpton 322, a Babylonian clay tablet from about 1800 BC, written in a sexagesimal number system.
context_section_description: There are 16 primitive Pythagorean triples with c ≤ 100:
  Note, for example, that (6, 8, 10) is not a primitive Pythagorean triple, as it is a multiple of (3, 4, 5).  Each of these low-c points forms one of the more easily recognizable radiating lines in the scatter plot.
  Additionally these are all the primitive Pythagorean triples with 100 < c ≤ 300:
caption: Scatter plot of the legs (a,b) of the first Pythagorean triples with a and b less than 6000. Negative values are included to illustrate the parabolic patterns. The "rays" are a result of the fact that if (a,b,c) is a Pythagorean triple, then so is (2a,2b,2c) and (3a,3b,3c) and, more generally, (sa,sb,sc) for any positive integer s.
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