Sunday, 28 March 2010

Perfect numbers

I can do no better than quote wikipedia:
In mathematics, a perfect number is a positive integer that is the sum of its proper positive divisors, that is, the sum of the positive divisors excluding the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself), or σ(n) = 2n. The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors, and 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 + 6 ) / 2 = 6.
'Perfect' is such a loaded term though. Why should we consider this neat trick of slicing a number into nonremaindering divisors and then recombining them a different way to arrive back at our starting place perfect? Diving 4 in half and multiplying the result by 2 is perfect, because it is a symmetrically circular process -- and because it works for all numbers. But the process of deriving divisors and recombing them as a sum is lopsided and doesn't work for all numbers.

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