Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

CHAPTER XV.
On the generation of the perfect number, and it$ similitude to
virtue.
ON account of the paucity, therefore, of perfect numbers,
there is only one between 1 and 10, viz. 6; one only between
10 and 100, viz. 28; between 100 and 1,000 only one, 496;
and between 1,000 and 10,000 the only perfect number is
8,128. These numbers likewise, are always terminated by the
two even numbers 6 and 8;" as is evident in those already adduced.
But the generation of them is fixed and firm, and can only
be effected in one way. For evenly-even numbers being disposed
in an orderly series from unity, the first must be added
to the second, and if a first and incomposite number is pre
duced by that addition, this number must be multiplied by the
second of the evenly-even numbers, and the product will be a
perfect number. If, however, a first and incomposite number
is not produced by the addition, but a composite and second
number, this must be passed by, and the number which follows
must be added. And if this aggregate is not found to be
a first and incomposite number, another must be added, and
this must be done till a first number is found. When therefore
this is found, it must be multiplied into the last of the added
evenly-even numbers, and the product will be a perfect number.
Thus for instance, in the evenly-even series of numbers
1. 2. 4. 8. 16. 32. 64. 128, if 1 is added to 2 the sum is 3, and
because 3 is a first and incomposite number, this multiplied
by 2 will produce the perfect number 6. But 28 the next per-
, Blxtius asserts, that perfect numbers always dtcma~ely end in 6 and 8; but
this is only true of the four first, and not of the rest