Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two 143
In the fourth place the greater the numbers are in the series
of perfectly and imperfectly amicable numbers the nearer they
approach to a perfect equality. Thus for instance, the exponent
of the ratio of 220 to 284 is 1& and by reduction 1%. But
1120
the exponent of the ratio of 17296 to 18416 is I-, and by
7 0 16
reduction I&. And rn is much less than E. Again,
the exponent of the ratio of 9363584 to 9437056 is 1-gh,
2 8 7 7 0
and by reduction I-. And is much less than i m.
In a similar manner it will be found that the exponents of the
ratios of the succeeding amicable numbers will continually decrease;
and consequently that the greater two amicable numbers
become, the nearer they approach to an equality with each
other. Indeed, in the amicable numbers after the first three,
this is obvious by merely inspecting the numbers themselves.
For in the amicable numbers 4827120640 and 4831837184, the
first two numbers from the left hand to the right are the same
in each, viz. 4 and 8. In the two amicable numbers
2473599180800 and 2473901154304, the first four numbers
2473 are the same in each. In the two amicable numbers
1266618067910656 and 1266637395132416, the first five numbers
are the same in each. This is also the case with the
two amicable numbers which immediately succeed these, viz.
648517109391294464 and 648518346340827136. And in the
two amicable numbers which are next but one to these
last, viz. in the numbers 170005188316757680455680 and
170005193383307194138624, the first seven numbers from the
left hand to the right are the same in each; by all which it
appears that the greater two amicable numbers are, the more
figures in the one are the same as those in the other, and consequently
that their approximation to a perfect equality is
greater.
The following are the remarkable properties of the prime
numbers and the numbers corresponding to prime, from
which both species of amicable numbers are produced.