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
Pythagorean prime Pythagorean prime A Pythagorean prime is a prime number of the form 4"n" + 1. Pythagorean primes are exactly the odd prime numbers that are the sum of two squares; this characterization is Fermat's theorem on sums of two squares. Equivalently, by the Pythagorean theorem, they are the odd prime numbers "p" for which is the length of the hypotenuse of a right triangle with integer legs, and they are also the prime numbers "p" for which "p" itself is the hypotenuse of a Pythagorean triangle. For instance, the number 5 is a Pythagorean prime; is the hypotenuse of a