#### That's perfect of course

As Euclid proved llong ago, if M(n) is a Mersenne prime, then the larget number P(n) given by (4^n - 2^n)/2 is a perfect number. Thus we have:

M(2) = 3, P(2) = (16 - 4) / 2 = 6

M(3) = 7, P(3) = (64 - 8) / 2 = 28

M(5) = 31, P(5) = (1024 - 32) / 2 = 496

M(7) = 127, P(7) = (16384 - 128) / 2 = 8128

These numbers are described as perfect because in each case the sum of factors is equal to the number itself, e.g. 28 = 1 + 2 + 4 + 7 + 14.

The new discovery of the 45th Mersenne prime therefore means ipso facto the discovery of the 45th known perfect number.

In later days Euler showed that Euclid's schema is the only schema for even perfect numbers. (Odd perfect numbers are an open question - it's known that there are none of 300 digits or less.)

(For the record, numbers where the sum of factors falls short of the number - the majority - are called "deficient", numbers where the sum of factors exceeds the number are known as "abundant". The first abundant number is 12. Odd abundant numbers are possible, the smallest being 945.)