PERFECT NUMBERS AND SOME ODDITIES 89

The perfect number 496 is the sum of the series 1 to 31,
and its various terms may be re-grouped as follows:
1. Qg g B TR U RS 3418 X0
31 30 29 28 27 26 25 24 23 22 21 20 19 18 17
Each of the first fifteen groups totals 32, leaving the num-
ber 16 unpaired. The total of all the terms is therefore
(15 X32) 416 or (15% X 32).
On the other hand, the factors of 496 are 1, 2, 4, 8, 16, 31,
62, 124, 248, and these can be re-grouped thus:
1431 =82 =1X42
2462 =64 =2Xx32
4+124=128=4X32
8+248=256=8 x 32
16 =10 =% X329

Total 151 X 32

In each case the number 496 is shown as being (15% X 32)
or 32232 1 other words, the last term of the progression

for a perfect number x must, when added to 1, give a number
n so that

 

 

n%—n
e
2
] n%—n
Now if ¥ s
2
then 2x=n?—n=n(n—1)

As x grows larger, so (n—1) becomes more nearly 7, so that
2x =n? approximately. Therefore the value of » may be found
as being very nearly equal to the square root of 2x.

For example, if x =496, the square root of twice this num-
ber is between g1 and 32, so that n=32, and (z—1) =31I.

The fact that these known perfect numbers are equal to
the sums of arithmetical progressions of the forms 1, 2, 3, 4,
etc., means that they must also be triangular numbers. They
are in fact particular triangular numbers, and it does not
follow that all triangular numbers are perfect.

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