14 
NEW METHOD OF FINDING THE 
with this question, we take it for granted that the number, 
whose factors are required, is odd. 
It is first necessary to show that every odd composite 
number is of the form — y^, that is, it is the difference 
between two squares (the proposition is general, but we need 
only here deal with odd composite numbers). 
Let a be a given odd composite number, and m and n 
a pair of its factors, so that a — ni x n. 
a being odd, all its factors are odd, so that m and n are 
odd. 
JLet — ^ — = X and — ^ — = y. 
2 2 ^ 
Note. — It is evident that both x and y are whole numbers, 
because, m and n being both odd, their sum and their differ- 
ence are both even. 
mi_ m -\- n . m — n 
Then — ^ — + = m = x + y, 
and — X_ — = w = x—y. 
2 2 
Therefore a = m x n =.{x y) {x—y) = — y"^. 
The problem of finding the two factors ?« and n, therefore, 
resolves itself into finding the integral values of x and y in 
the following indeterminate equation, namely, 
a = x^—f, 
■which we may wi-ite in the more convenient form of 
y- =. x"- — a. 
In other words, we have to find a square number ar, 
such that, if, after subtracting fi-om it the given number a, 
we get a square number for a remainder, then this remainder 
will equal y'^ ; and one integral solution of our equation will 
have been found. The factors ■v\ill then be x + y and x — y. 
There is, I believe, no dii'sct way of solving tliis equation, 
but the following method will enable us to arrive at the 
solution without any large amount of labor. 
