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
Legendre's three-square theorem Legendre's three-square theorem In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers if and only if is not of the form formula_2 for integers and . The first numbers that cannot be expressed as the sum of three squares (i.e. numbers that can be expressed as formula_2) are Pierre de Fermat gave a criterion for numbers of the form 3"a" + 1 to be a sum of three squares essentially equivalent to Legendre's theorem, but did not provide a proof. N. Beguelin noticed in 1774 that every positive