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24 THE FASCINATION OF NUMBERS

It should be noted that if =2 and b=4 then 245 is still 16
as before, but a different set of figures will emerge. In this
case x*+92=22, becomes 62+82=102. There is in fact a dif-
ferent set of numbers for each different factorization of 2ab.

For example, if 2ab =64, then ab=3a2.

and a=1; b=3g2
or a=2; b=16
or a=4; b=8

Any of these sets substituted in the formula given will result
in a set of Pythagorean numbers.

If 32+42=5?2; then it is obvious that 62+82=r102,

For if 3%+4%=5?

then (22)(32+42) =(2)2(5)®
and (22X 32) 4 (22 X 42%) =22 x 52

and (2 X3)2+(2 X4)2=(2 x52)
and 62+82=10?

The difference between two consecutive square numbers is
equal to the sum of their square roots.
Thus, 722 —71%2=724-71=143

This is because, if we follow the algebraical equation
%2 —y?=(x-y)(x—) we have 722—712=(72+71)(72—71)
and since we have chosen two consecutive numbers then
x—y (or 72—71 as here) is always 1, so that this bracket of
the factorization can be ignored.

All square numbers are, if even, a multiple of 4 or, if odd,
1 more than a multiple of 4. This is obvious. Every even
number is of the form 2n, and the square of this is 4n? which
is plainly a multiple of 4. On the other hand, every odd num-
ber is of the form 2741, and the square of this is 4n2+4n-+1.
This is 1 greater than 4n2 - 4n, which is exactly divisible by 4.
But if z is odd, then #24-7 is even and 4n®-+-4n becomes a
multiple of 8. It follows that odd squares are really 1 more
than a multiple of 8.

The fact that the total of any series of consecutive odd