A perfect number is a counting 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 $2^{p-1}(2^p-1)$ where $2^{p-1}$ is a prime number.

This formula was first proved by Euclid.

There are 47 known perfect numbers of the form $2^{p-1}(2^p-1)$ 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 $2^{43,112,608}(2^{43,112,609}-1)$ has 25,956,377 digits.

Primes of the form $2^{p-1}$ 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

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

• Show that every even perfect number is of the form $2^{p-1}(2^p-1)$ where $2^p-1$ is prime.

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