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88 THE FASCINATION OF NUMBERS

The first five perfect numbers can now be shown:

6 =Sum of progression 1 to 3  inclusive

28 = 9 ”» Ito 7 %)
4962 391y Y) I to 31 »
BIoBes il % I to 127 "

33:5505336= 29 iy » I to 8IQI %)

The last term is, in every case, a prime (being of the same
form as in Euclid’s formula), and the only other factors of the
perfect numbers are 1 and varying powers of the number 2.

The relationship between each progression is as follows.
Once the last term (/;) of an earlier progression is known, the
last term (/;) of the next progression is found by multiplying
[, by 2 and adding 1. So, /,=2(/;) 4 1; but only provided the
resulting /, is prime. For example, as between the two num-
bers 6 and 28, the last terms of their respective progressions
are 3 and 7 (thatis, /; =3; [,=7; and therefore l,=2(1;) +1).
If, however, the resulting /, is not a prime then the pro-
cedure of doubling and adding 1 (at each stage) is continued
until a prime does arise. This prime, when reached, will then
be the last term of the next perfect number’s progression.

This explains the gap between the fourth and fifth perfect
numbers. The last term of the progression for the fourth
number is 127. The successive doubling up and addition of
unity at each stage give the following:

(Fourth number) Last term of A.P.= 127 Prime
Double and add 1= 255 Not Prime

Y » sy R M 3 3
n ” s 1=1023 » 2
” » s 1=2047 %) ”»
” ) s 1=4095 3 3
. s 3 (I==8101. Prime

The last term of each progression also bears a direct rela-
tionship to its perfect number other than as a mere factor,
and it is instructive to compare the structure of a perfect
number (regarded as the sum of an arithmetical progres-
sion) with its factorial composition.