Chapter 4: Systems Based on Numbers

Section 4: The Sequence of Primes

Perfect numbers

Perfect numbers with the property that Apply[Plus, Divisors[n]]==2n have been studied since at least the time of Pythagoras around 500 BC. The first few perfect numbers are {6, 28, 496, 8128, 33550336} (a total of 39 are currently known). It was shown by Euclid in 300 BC that 2n-1(2n-1) is a perfect number whenever 2n-1 is prime. Leonhard Euler then proved around 1780 that every even perfect number must have this form. The values of n for the known Mersenne primes 2n-1 are shown below. These values can be found using the so-called Lucas-Lehmer test Nest[Mod[#2 - 2, 2n-1] &, 4, n - 2] == 0, and in all cases n itself must be prime.

Perfect numbers image 1

Whether any odd perfect numbers exist is probably the single oldest unsolved problem in mathematics. It is known that any odd perfect number must be greater than 10300, must have a factor of at least 106, and must be less than 44s if it has only s prime factors. Looking at curve (b) on page 135, however, it does not seem inconceivable that an odd perfect number could exist. For odd n up to 500 million the only values near 0 that appear in the curve are {-6,-5,-4,-2,-1, 6,18,26,30,36}, with, for example, the first 6 occurring at n=8925 and last 18 occurring at n=159030135. Various generalizations of perfect numbers have been considered, requiring for example IntegerQ[DivisorSigma[1,n]/n] (pluperfect) or Abs[DivisorSigma[1,n]-2n] < r (quasiperfect).

From Stephen Wolfram: A New Kind of Science [citation]