perfect numbers

It is not known when perfect numbers were first studied and indeed the first studies may go back to the earliest times when numbers first aroused curiosity. It is quite likely, although not certain, that the Egyptians would have come across such numbers naturally given the way their methods of calculation worked. Today the usual definition of a perfect number is in terms of its divisors, but early definitions were in terms of the 'aliquot …

…is equal to: 2 ^ (n-1) * ( 2^n - 1) where n = 2, 3, 5, 7, 13, 17, 19, 31 or other "mersenne" exponents. All of the known "Mersenne" exponents are listed in one of the sections above. Can we prove that every "Mersenne Prime Number" can generate a corresponding Perfect number? Yes, we can. What do you know? That's the perfect number we started with. So we can see that 2 ^ (n-1) * ( 2^n - 1 ) will always be a perfect number whenever (2^n - 1) is a prime number.