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Pythagoras of Samos (Greek: Πυθαγόρας; between 580 and 572 BC–between 500 and 490 BC) was an Ionian (Greek) philosopher^[1]
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Pythagoreanism is a term used for the esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were much influenced by mathematics and probably a main inspirational source for Plato and platonism.
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In mathematics, the Pythagorean theorem (AmE) or Pythagoras' theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. The theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and
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Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c².
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A Pythagorean prime is prime number of the form 4n + 1. These are exactly the primes that can be the hypotenuse of a Pythagorean triangle.

The first few Pythagorean primes are

5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113

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The Pythagorean trigonometric identity is a trigonometric identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae (see trigonometric identity#Angle sum and difference identities) it is the basic relation among the
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Pythagorean comma, named after the ancient mathematician and philosopher Pythagoras. It is sometimes called a ditonic comma.

Put more succinctly, twelve perfect fifths are not exactly equal to seven perfect octaves, and the Pythagorean comma is the amount of the
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Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. Its name comes from medieval texts which attribute its discovery to Pythagoras, but its use has been documented as long ago as 3500 B.C.
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According to legend, Pythagoras discovered the foundations of music by listening to the sounds of four blacksmith's hammers, which produced consonance and dissonance when they were struck simultaneously.
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Pythagorean expectation is a formula invented by Bill James to estimate how many games a baseball team "should" have won based on the number of runs they scored and allowed.
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