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62 THE FASCINATION OF NUMBERS

Since a is always odd, (a—1) or m must always be
even. Tests reveal:

9=32

15=3%+2(3)

21 =3%2+4(3)

25 =52

27=3%+6(3) =52+2(1)
33=32+8(3) = 2+8(I)
35=5%+2(5)
39=3%+10(3) =5%+14(1)

45=3"+12(3)

In each case C is either a perfect odd square; or a perfect
odd square plus an even multiple of that square’s root. But
what is of much greater significance is the fact that that
root (n) is a factor of C, and that (n-+m) is another factor.

Thus where C=2ay
=3
and n+m=g
This relationship is easily proved since
C=n(a+n—1)
=n(n+m)

The procedure for the use of this method is as follows:
Let C=11,111

Nearest odd square less than C=11,025 (=1052)
11,111 =(1052)+86
The remainder 86 is not a multiple of 105, so that 105 is
not a factor of C. A process of elimination, trying succes-
sively lower odd squares at each stage, eventually leads to:

11,111=(41)%+9430
=(41)*+230(41)
The factors of 11,111 are therefore 41 and 271 (=41 +230).
By now it is clear that the series of consecutive odd num-
bers is of immense importance in the theory of numbers, and
it is significant to note that these numbers are themselves