Pythagorean

Pythagorean refers to anything relating to Pythagoras of Samos, his life and philosophy, or to the mathematical principles attributed to him. The term can describe a person who adheres to Pythagoreanism, a historical philosophical and religious movement in the ancient Greek world, largely associated with the community founded by Pythagoras in Magna Graecia during the 6th century BCE. The Pythagoreans emphasized numbers, harmony, and the belief in the transmigration of souls, and their ideas influenced later mathematics, philosophy, and science.

In mathematics, the adjective Pythagorean is commonly linked to the Pythagorean theorem, which states that in a right triangle the square of the hypotenuse equals the sum of the squares of the other two sides. Although associated with Pythagoras and his followers, proofs of the theorem appear in later classical geometry as well. The term also covers Pythagorean triples, integer solutions to a^2 + b^2 = c^2, such as (3,4,5) and (5,12,13). The Pythagoreans contributed to the study of number theory, including investigations into perfect numbers, amicable numbers, and the connections between numbers and musical pitch.

Beyond antiquity, the word Pythagorean can describe schools, groups, or movements inspired by Pythagoras’s teachings, including historical communities and some modern organizations that emphasize ethical, mathematical, or philosophical ideas. In scholarly usage, distinctions are drawn among Pythagoras as a historical figure, the Pythagorean school or movement, and Pythagoreanism as a broader worldview.