A positive integer is called perfect provided the sum of its
proper positive divisors is x. For example, 6 is perfect because 6
= 1 + 2 + 3: Another perfect number is 496. Indeed, 496 = 31 16,
and we see that the positive proper divisors of 496 are 1; 2; 4; 8;
16; 31; 31 2 = 62; 31 4 = 124; 31 8 = 248: Then we add them up and
get 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496: If we include x
as divisor of itself, we see that x is perfect if and only if the
sum of all of its positive divisors (including x) is 2x.
Question: A number of the type 2^n -1 is sometimes a prime, then
called a Mersenne prime. If 2^n -1 is a prime, show that x =
2^(n-1)(2^n -1) is a perfect number.
Hint. You should be able to see a pattern in the factors of x
and also to add them up. Don’t forget that 2^n -1 is given to be
prime.