Pythagorean triple

A Pythagorean triple consists of three positive integers $a$ , $b$ , and $c$ , such that $a 2 + b 2 = c 2$ . Such a triple is commonly written $(a, b, c)$ , 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 triangle whose side lengths are a Pythagorean triple is a right triangle and called a Pythagorean triangle.

A primitive Pythagorean triple is one in which $a$ , $b$ and $c$ are coprime (that is, they have no common divisor larger than 1).^[1] For example, $(3, 4, 5)$ is a primitive Pythagorean triple whereas $(6, 8, 10)$ is not. Every Pythagorean triple can be scaled to a unique primitive Pythagorean triple by dividing $(a, b, c)$ by their greatest common divisor. Conversely, every Pythagorean triple can be obtained by multiplying the elements of a primitive Pythagorean triple by a positive integer (the same for the three elements).

The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula $a^{2}+b^{2}=c^{2}$ ; 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 $c={\sqrt {2}}$ is a right triangle, but $(1,1,{\sqrt {2}})$ is not a Pythagorean triple because ${\sqrt {2}}$ is not an integer. Moreover, $1$ and ${\sqrt {2}}$ do not have an integer common multiple because ${\sqrt {2}}$ 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. It was discovered by Edgar James Banks shortly after 1900, and sold to George Arthur Plimpton in 1922, for $10 (equivalent to $182 now).^[2]^[3] Another ancient reference to Pythagorean triples is found in Baudhayana (One of the four Shulvasutras).

When searching for integer solutions, the equation $a 2 + b 2 = c 2$ is a Diophantine equation. Thus Pythagorean triples are among the oldest known solutions of a nonlinear Diophantine equation.

^ Long (1972, p. 48)
^ 1634–1699: McCusker, J. J. (1997). How Much Is That in Real Money? A Historical Price Index for Use as a Deflator of Money Values in the Economy of the United States: Addenda et Corrigenda (PDF). American Antiquarian Society. 1700–1799: McCusker, J. J. (1992). How Much Is That in Real Money? A Historical Price Index for Use as a Deflator of Money Values in the Economy of the United States (PDF). American Antiquarian Society. 1800–present: Federal Reserve Bank of Minneapolis. "Consumer Price Index (estimate) 1800–". Retrieved February 29, 2024.
^ Robson, Eleanor (2002), "Words and Pictures: New Light on Plimpton 322" (PDF), The American Mathematical Monthly, 109 (2): 105–120, doi:10.1080/00029890.2002.11919845, S2CID 33907668

[1] Long (1972, p. 48)

[inflation-US-2] 1634–1699: McCusker, J. J. (1997). How Much Is That in Real Money? A Historical Price Index for Use as a Deflator of Money Values in the Economy of the United States: Addenda et Corrigenda (PDF). American Antiquarian Society. 1700–1799: McCusker, J. J. (1992). How Much Is That in Real Money? A Historical Price Index for Use as a Deflator of Money Values in the Economy of the United States (PDF). American Antiquarian Society. 1800–present: Federal Reserve Bank of Minneapolis. "Consumer Price Index (estimate) 1800–". Retrieved February 29, 2024.

[3] Robson, Eleanor (2002), "Words and Pictures: New Light on Plimpton 322" (PDF), The American Mathematical Monthly, 109 (2): 105–120, doi:10.1080/00029890.2002.11919845, S2CID 33907668

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