Mathematical_Recreations-Kraitchik-2e

Mathematical Recreations
and 28 = 1 + 2 + 4 + 7 + 14. The only perfect numbers
known are even numbers of the form 2,,-1 (2" - 1) in which
the second factor is prime. Only the following perfect
numbers are known, those obtained by setting n = 2,3,5,
7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281.
Concerning these numbers Mersenne wrote: "On voit
clairement par la combien sont rares les N ombres Parfaits et
combien on a raison de les comparer aux hommes parfaits."
["We see clearly from this fact how rare are Perfect Numbers
and how right we are to compare them with perfect
men."]
We are in a century of materialism, and in the opinion of
our time, numbers, however marvelous their properties, possess
no human qualities. But in Mersenne's time everyone
believed in the intrinsic qualities of numbers, which were not
only perfect, but also lucky or unlucky, good or bad, and so
forth; so Mersenne's statement is not strange.
Mr. Heath has proved that every perfect number
2,,-1(2" - 1) is a sum of cubes of 2k odd numbers, when
k = !en - 1), except for n = 2. The proof is as follows:
Recall that
m(m + 1)
8 1 = 1 + 2 + 3 + ... + m = 2 '
8 2 = }2 + 22 + 32 + ... + m2 = m(m + 1)(2m + 1)
6 '
Then
8 }3 + 33 + 53 + ... + (2m - 1)3 = t(2m - 1)3
i-I
t(8m3 - 12m2 + 6m - 1)
i-I