13 
On a New Method of finding the Factors 
of any given Number: 
A contribution to the Theory of Numbers. 
(Bv G. K. WINTEE, M.I.C.E., F.R.A.S.) 
In the first volume of Stanley Jevons' Principles of Science, 
page 141, the author gives a pretty illustration of the greater 
difficulty of inverse processes as compared with direct ones, 
and this illustration is given in the following words : — 
' ' The same difficulty arises in many scientific processes. 
Given any two numbers, we may by a simple and infallible 
process obtain their product, but it is quite another matter ■when 
a large number is given to determine its factors. Can the reader 
gay what two numbers multiplied together will produce the 
number 8,616,460,799 ? I think it unlikely that any one but 
myself will ever know, for they are two large prime numbers, 
and can only be rediscovered by trying in succession a long 
series of prime divisors until the right one be fallen upon. The 
work would probably occupy a good computer for many weeks, 
but it did not occupy me many minutes to multiply the two 
factors together." 
The secret of these numbers probably died with him, and 
their rediscovery possesses a certain interest, which will be 
readily appreciated. Apart from this, however, the resolu- 
tion of a number into factors by any means other than by 
direct trial, has, as far as I am aware, not hitherto been done, 
and, if I am right in this, I think the method by means 
of which these numbers have been rediscovered may possess 
some theoretical value. 
Every number is either odd or may be reduced to an odd 
number by repeated division by two, and therefore, in dealing 
