9
Perfect Numbers and Some Oddities

There are certain numbers which individually or as mem-
bers of a group possess properties peculiar to themselves.
Some of these are related to each other, but others have
no apparent relation to any other numbers with similar
properties.

The easiest to classify are the perfect numbers. A perfect
number is one which is itself equal to the sum of all its fac-
tors (including 1 as a factor). The lowest of such numbers is
6, whose factors (1, 2 and g) also total 6. Only twelve of
these numbers are known. The first four are 6, 28, 496 and
8128; the fifth is 33,550,336 and the others are much greater.
All of these known numbers end either with the digit 6 or
the two digits 28; and so far no odd perfect numbers have
been discovered, unless we accept 1 as such.

Euclid proved that any number of the form (27) (2"*1—1)
is a perfect number when the factor (2"*1—1) is a prime
number. Thus 496 can be factorized to (2%) (2°—1) or
(16) x (31), and since 31 is a prime, then 496 is perfect. It will
be observed that the known perfect numbers become much
less frequent as numbers grow larger. This is because the
primes on which they are based become more rare.

There is a way of building up perfect numbers without
having direct regard to Euclid’s formula. The factors of
6 are 1, 2 and 3; and these are in the form of an arithmetical
progression. This provides a clue that the other perfect num-
bers may also prove to be equivalent to the sums of arith-
metical progressions. This is, in fact, the case; and each of
these progressions is inter-related to the others.

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