Pythagoreanism to have survived as part of, or deeply influenced, the Bacchic cults and Orphism. Pythagoras was already in ancient times well known for the mathematical achievement of the Pythagorean theorem. Although the Babylonians had used this formula for centuries, Pythagoras had discovered that "in a right-angled triangle the square of the hypotenuse is equal to the squares of the other two sides". In ancient times Pythagoras was also noted for his discovery that music had mathematical foundations. Antique sources that credit Pythagoras as the philosopher who first discovered music intervals also credit him as the inventor of the monochord, a
Fermat's right triangle theorem single Diophantine equation which can be simplified to Conversely, any solution to this equation could be factored to give a square congruum. (In particular, the squares formula_13, formula_14, and formula_15 form an arithmetic progression with congruum formula_16, which is a square itself.) Thus, the solvability of this equation is equivalent to the existence of a square congruum. But, if Fermat's last theorem were false for the exponent formula_17, then squaring one of the three numbers in any counterexample would also give three numbers that solve this equation. Therefore, Fermat's proof that no Pythagorean triangle has a square area implies that