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30 THE FASCINATION OF NUMBERS

Cube numbers can also be represented as the difference
between the squares of consecutive triangular numbers.

Since 13423433 =(14243)2

and 13423 =(142)?2
then 33=(1+2+3)2—(1+2)*
:62_32

This explains why, for each cube number, only certain con-
secutive terms of the series of odd numbers are included.

Another structure of the cube numbers is revealed by the
fact that each such number is a multiple of 7 or differs from
a multiple of 7 by 1. That is, each cube number is of the form
7n or 7n41. Therefore no number can be a cube number if it
leaves a remainder of 2, g, 4 or 5 when divided by 7.

Any ‘power’ number (square, cube, etc.) can be repre-
sented as the sum of a particular arithmetical progression
and itself forms part of a series in its own particular power
category. There is an interesting relationship between the
actual series which form the various ‘power’ numbers. This
can be seen more clearly if the various series are written out
and reduced to progressions having a constant difference
between terms.

SQUARES
I 4 9 16 25 36
8 5 7 9 I
2 2 2 2
(Thus after one step, the series is reduced to a progression with
a common difference of 2.) .
CUBES
I 8 27 64 125 216
7 19 37 61 91
12 18 24 30

6 6 6