54 THE FASCINATION OF NUMBERS

included. Where all the basic factors have the index 1, the
sum of all the different factors is found by adding 1 to each
of the basic factors and multiplying them together. Where,
however, any of the basic factors, a, b, ¢, etc., have an index
greater than 1, as in the number a%6%" . . . , then the sum of
all the different factors is given by

£4p+1_ 1) (B9l —1) (f+1—1)
(@a—1) (b—1) (¢—1)
The basic factors of the number 60 are 22.3.5. The sum of
all the different factors is therefore

(22—1) (3*—1) (5°=1) __ .o

(2—1) (3—1){5~—1)
This can easily be checked by simple addition, the factors
being 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 6o.

Fermat was responsible for the clarification of many prin-
ciples affecting number theory, but his expressed opinion
that all numbers of the form 2% 41, where x=2", are primes,
was subsequently proved by Euler to be incorrect. Numbers
of this form are called Fermat Numbers for obvious reasons.
Where n=p5, the expression 22" +1=4,294,967,297 and this
number has the factors 641 and 6,700,417.

In 1644, the mathematician Mersenne suggested that in
order that numbers of the form 2 —1 should be prime, the
only possible values of P, not greater than 257, were 1, 2, 3,
5 7> 13, 17, 19, 31, 67, 127 and 257, although the number
67 was probably a misprint for 61. In point of fact, not all of
these values make 2 —1 prime. Where P=257, the expres-
sion is composite.

Two other values of P (89 and 107) have been proved to
generate prime numbers. Mersenne’s Numbers are closely
related to the formula for generating perfect numbers, to
which reference is made in a later chapter. For 75 years, the
value P=127 gave the largest prime number then known:

=170,141183,460469,231731,687303,715884,105727