Saturday, 4 June 2011

Perfect numbers

A number is said to be perfect if it is equal to the sum of all its divisors, excluding itself.  Perfect numbers have been studied by ancient Greek mathematicians including Euclid.  They are closely related to Mersenne primesEuler has proved that all even perfect numbers can be written as:

for all prime p associated with a Mersenne prime.  The last expression above hints that even perfect numbers must be triangular.  The first few even perfect numbers, when p = 2, 3, 5, 7, 13, are:

6, 28, 496, 8128, 33550336.

It is not known whether odd perfect numbers exist.

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