## Perfect Numbers

A perfect number is equal to the sum of it's 'proper divisors', that is, all number less than the number itself. Thus 6 is a perfect number because the divisors of 6 are 1, 2, 3 and 1+2+3=6.

The next three perfect numbers are:

28 = 1 + 2 + 4 + 7 +14

496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248

8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064

Perfect numbers are very rare. At the moment only 29 are known. Perfect numbers are believed to be all even – no odd perfect numbers are know, and if they exist they must be larger thanAll even perfect numbers may be generated by the following expression,:(1) where bothandare prime numbers.

Proof:

The proper divisors ofareand

The sum of the proper divisors is

is the sum of a geometric series with n terms, first therm 1 and common ratio 2. Henceand

Hence the sum of the proper divisors is

Euclid discovered the expression (1). Euler later proved that all even perfect numbers are of this form.

The numberswhich occur in (1) have a special name – Mersenne primes – and a special symbol,through in factis not always prime, because of course, no formula exists for generating only prime numbers. If p=137, 139, 149, 199, 227, 257is not prime.

The perfect numbers have the following additional properties:All perfect numbers end in 6 and 8 alternately.There are infinitely many perfect numbers.

The reciprocals of the divisors of a perfect number N must add up to 2:

• For 6, we have 1 / 6 + 1 / 3 + 1 / 2 + 1 / 1 = 2;

• For 28, we have 1 / 28 + 1 / 14 + 1 / 7 + 1 / 4 + 1 / 2 + 1 / 1 = 2, etc.

Every even perfect number, other than 6, is the sum of consecutive odd cubes.

All even perfect numbers are triangular numbers.

etc.