Fun with Numbers #17 – Perfect numbers

A perfect number is a number n where if all the factors of n were summed up and divided by 2 then the result is n. An example of this is the first perfect number 6, the factors of 6 are 1, 2, 3 and 6: (1+2+3+6/2) = 6. The first four perfect numbers 6, 28, 486 and 8128 were known and discovered by the classical Greeks, the 5th number was discovered in the middle ages the and the 8th was discovered by Euler.

Euler also strengthened an observation that Euclid made about the Perfect numbers. Euclid had noticed that given a perfect number, it can be expressed as p(p+1) where p is a Mersenne Prime (a prime number that can be expressed as 2^n+1). Euler expanded on this by showing that p(p+1), where p is a Mersenne prime will always result in a perfect number, thus each perfect number will correspond to a Mersenne prime. 6 and 3, 28 and 7, 486 and 31 and so on.

As a result there are as many Perfect numbers as there are Mersenne Primes, however whether or not there is a finite number of Perfect numbers is still unknown.

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