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HEADERS_END

A perfect number is a number which is the sum of all its divisors (apart from itself).

Examples:

* 6 = 1+2+3

* 28 = 1+2+4+7+14

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

These are all of the form EQN:2^{p-1}(2^p-1) where EQN:2^{p-1} is a prime number.

This formula was first proved by Euclid.

There are 47 known perfect numbers of the form EQN:2^{p-1}(2^p-1) when p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609.

The largest EQN:2^{43,112,608}(2^{43,112,609}-1) has 25,956,377 digits.

Primes of the form EQN:2^{p-1} are called Mersenne Prime numbers.

All even perfect numbers are of the form show above however it is an open conjecture as to whether there are any odd perfect numbers.

It has been proven that any odd perfect number must have at least 47 prime factors !!!

“Perfect numbers like perfect men are very rare”. Rene Descartes 1596 - 1650

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Enrichment Task

Show that if EQN:2^p-1 is a prime number then EQN:2^{p-1}(2^p-1) is a perfect number.

Much harder: Show that every even perfect number is of the form EQN:2^{p-1}(2^p-1) where EQN:2^p-1 is prime.