20 NEW METHOD OF FINDING THE FACTORS OF ANY NUMBER. 
49 — 45 is 4, whicli we see at once is a square number. We 
know, therefore, that 7 is the middle, and that ^4 = 2 is the 
difference between the middle number and the factors ; thus one 
factor is 7 + 2 = 9 and the other is 7 — 2 == 5. In the above 
case the difference 4 between a and ¥ was a perfect square, and 
the solution was found at once ; but now let « = 39 then V — a 
= 10 which is not a square number. The middle number must 
be greater than 7. Take 8 the next number. The square of 8 
is 64, and 64 — 39 = 25, which is a square number, therefore 8 
is the middle number, and the difference between the middle 
number and each factors is /\/25 = 5 ; the factors are therefore 
8 + 5 = 13 and 8 — 5=3. 
It has been pointed out to me that every odd number 
may be expressed as the difference between two squares. 
This is of course quite true, for a = — 0 ~2 ^ ) ' 
This merely indicates that one pair of factors is a and 1, 
In the order in which the factors are found by the process 
described, this solution would be found last ; and, if no other 
solution be found, the number must be prime. This con- 
sideration shows that the operation need not be carried 
beyond finding the difference — ^• 
If a table of squares is available, the actual subtractions 
may be performed, rejecting in the process all those squares 
which have the last two figures such that, after the sub- 
traction, the differences could not be squares. The process 
would in this way be somewhat similar to that used for 
testing numbers of the form 4« + 1 for primes. Such 
numbers are prime when they are the sum of two squares, 
but only of two. 
