‘PRIMES AND FACTORS’ 61

whence a=7%3, and n=g9, so that the number 729 can also be
expressed as the sum of nine terms of which the first term
is 73:
73+75+77+79+81+83+85+87+89=729

Yet another entirely new test of primality may be de-
veloped from the series of consecutive odd or even numbers.
It is much simpler to explain and to employ than is the first
method shown above; it is also more practical and therefore
of much greater importance. Theoretically it provides a
certain method of testing any number, although the greater
the number so also the longer the process. It also reveals
what, if any, are the factors.

The method may be stated briefly thus:

(a) If a Composite number is built up from an even series,
it must itself be even and is thus easily recognizable
as composite.

(b) If a Composite number, C, is built up from an odd
series, it must be of the form of an arithmetical pro-
gression

C=3%(n)(a+1)
But where the common difference is 2

l=a+2(n—1)
Therefore: j

1(n)(a+!) =%4n(a+a-+2n—2)

=n(a+n—1)=C

If we substitute values for a

a=1 C=n?®

a=3 C=n2-+2n

a=5s C=n%+44n

a=9g C=n2+8n
This may be expressed in the general formula

C=n2+mn

where mn is some even multiple of n, and m=(a—1).

R R R

T

GUR I

i . A L s RS LTI L G 0 e Esn e
MWW“WMHMMMMMMMA R ABBRNANTANOGHRIRE

R TR s TS

RGN

70

CUE GRS
EMEREIREA