: Matemáticas: Pythagorean triple

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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 $(a, b, c)$ , 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 (that is, they have no common divisor larger than 1). For example, $(3, 4, 5)$ is a primitive Pythagorean triple whereas $(6, 8, 10)$ is not. 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 ${\\displaystyle a^{2}+b^{2}=c^{2}}$ ; 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 ${\\displaystyle a=b=1}$ and ${\\displaystyle c={\\sqrt {2}}}$ is a right triangle, but ${\\displaystyle (1,1,{\\sqrt {2}})}$ is not a Pythagorean triple because ${\\displaystyle {\\sqrt {2}}}$ is not an integer. Moreover, ${\\displaystyle 1}$ and ${\\displaystyle {\\sqrt {2}}}$ do not have an integer common multiple because ${\\displaystyle {\\sqrt {2}}}$ 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.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Pythagorean_triple)