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56 THE FASCINATION OF NUMBERS

series; there is instead a pair of numbers and these are
respectivelv 1 less and 1 greater than n®-1. For 44, the central
pair of numbers are 63 and 65 (that is, 64—1 and 64+1)
respectively, so that:

4*=61-+63+65+67=256
There is a simple reason why this relationship should exist.

If an odd number has a factor, then if it be divided by that
factor, we can represent it as the sum of its parts, thus:

93=31+3I+3I
If we deduct 2 from the first of these parts and add 2 to the
last, we obtain:

93=29+3I+33
which is obviously the same.

In the same way, any even number may be divided into

equal parts:

64=16416416416
This time we deduct g from the first term and add 3 to the
last term; and deduct 1 from the second term and add 1 to
the third term, so that:

64=13+15+17+19
This property of divisibility and convertibility to terms of
odd numbers is possessed by most composite numbers. An
important fact which emerges, however, is that no prime
number can be treated in the same way, for the simple reason
that they have no factors other than themselves and unity.
This suggests a new way for identifying primes.

Let us first examine those numbers which can be built up
from the odd number series. We find that all these are com-
posite numbers. For the numbers 1 to 50, we obtain the fol-
lowing results:

Derived from the odd series

1, 4, 8, 9, 12, 15, 16, 20, 21, 24, 25, 27, 28, 32, 33, 35,
36, 39, 40, 44, 45, 48, 49