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
Automedian triangle always odd, all the sides "a, b, c" have to be odd. This fact allows automedian triples to have sides and perimeter of prime numbers only. For example, (13, 17, 7) has perimeter 37. Because in a primitive automedian triangle side "a" is the sum of two squares and equal to the hypotenuse of the generating primitive Pythagorean triple, it is divisible only by primes congruent to 1 (mod 4). Consequently, "a" must be congruent to 1 (mod 4). Similarly, because the sides are related by formula_18, each of the sides "b" and "c" in the primitive automedian is the