 This definition is ancient, appearing as early as Euclid's Elements (VII.22) where it is called ("perfect", "ideal", or "complete number").Euclid also proved a formation rule (IX.36) whereby formula_1 is an even perfect number whenever formula_2 is a prime of the form formula_3 for prime formula_4—what is now called a Mersenne prime.Two millennia later, Euler proved that all even perfect numbers are of this form.