18 
NEW METHOD OF FINDING THE 
1876426 This is the difference between the given number 
2062097 
185673 
Do. 
do. 
928362 
2247770 
185675 
Do. 
do. 
928372 
2433445 
185677 
Do. 
do. 
92838* 
2619122 
Do. 
do. 
928392 
* 
# 
* 
and so on, the number added being increased by 2 each time. 
The possible squares, of which there are only two in the 
above fifteen diffei-ences, are marked *. 
In this way it is found that the first difference which is 
a complete square is the number 10233601, and its square 
root is 3199. This last number added to and subtracted 
from 92880 (which is the number whose square, minus the 
given number, is the square number 10,233,601, found as 
above) will give the numbers 96,079 and 89,681, which are 
the required factors. 
Having found two factors as above, we can go on and 
test for others in the same way ; but it will be foimd easier 
to treat each of the factors themselves in the same way as 
we have treated the original niunber. In the present case, 
however, the factors 96,079 and 89,681 we know are prime 
numbers, and fiu'ther trial is of course unnecessary. 
The following reasoning, which first led me to the method 
described, may be interesting : — 
Let us suppose that the number we are dealing with is 
odd. Then all its factors must be odd, and therefore between 
any pair of its factors there must be a middle integral 
number. Let this be .v for a certain paii' of factors, then 
