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48 THE FASCINATION OF NUMBERS

higher numbers for ¢ until we reach the point where

(a+x)%—Nis a perfect square (52). To simplify this, we may

substitute y for (a+=x) and show the equation y2 — N =42,
From this equation we have:

—.N'zbz
—b2=N
= (=b) v+b =N

and the factors of .N are therefore (y—b) and (y+b).
This may best be illustrated by an example, showing the
various stages:

Detail Numerical example

Number to be

factorized: N=323
Next square

greater than N: a®=324 (=182
Difference: a®—N=b%?=324—323=1
Thus: a® —N=05b2=182—g23=12
and: a*—b*= N =182—1%=323
and (a+0b) (a—b)=N=(18+1) (18 —1)=323

and this gives 19 and 17 as the factors of 323.

In this example the difference 182—323 was a perfect
square. If, however, it had not been a perfect square, then
we would have formed in turn (192—323), then (202 —323)
and so on until a perfect square did result or was proved to
be impossible. It would not, of course, be necessary to calcu-
late the numerical value of each of these identities separate ly
since each square number can be obtained from the previous
one by virtue of the relationship shown in Chapter 2. There
it was seen that 182 was the equivalent of the sum of the first
eighteen terms in the series of odd consecutive numbers
1,3, 5 7, ...and that the eighteenth term of the series was
equal to (2 X 18) —1=35. In order to obtain the value of 192
we have only to add the next term in the series (37) to 182,

Thus: 1824-37=361=192

It follows, therefore, that instead of squaring each number