(432 ) 
then by the operation the other number would have been found, 
and not 539. 
As 
716 716 716 716 
== 102, +4 ii = 65.03 4 =De 
it is immediately evident, that 953 cannot contain a factor smaller 
than 14, or between 49 and 65, or between 77 and 102. 
The first limit 14 shows that 13 cannot be a factor. 
From the last limits would ensue, that there are no factors between 
955 953 
El Amd 
65 102 
The possibility that 953 is divisible by 7 or 11 (which would 
bring about no change in the result 539 > 953) is excluded with 
respect to 11, because 11 lies between 9 and 14; so that if by 
direct division 7 proves to be a non-divisor of 953, the factors of 
7, 11 and 13 are excluded, and so 17 would have to come under 
consideration as lowest factor. 
So the last operation in § I (pag. 331) has been continued already 
too far. For after the second addition we get: 
== By ues 
953 = 372 — 416 — 37? — (20,...)?, 
so that the difference 17 is already exceeded. 
The preceding can be applied only to one of the factors if the 
other one has been factorized. The following extension can however 
be given : 
Example: G == 8695261 is obtained of 9803 < 887, thus equal 
to 5345? — 44582, 
As /G is equal to 2948,... the operation according to the method 
of § I would be rather long. 
Suppose we know that one factor is more than 10 times but less 
than 14 times the other factor, then by multiplying G by 11, 12 
or 13 we can obtain a number that can be decomposed into two 
factors differing but slightly. 
In this way 11 x G would give: Gj = 95647871=9803 x9757= 
97802—46?, and the extraction of the root would immediately furnish 
a result. 
Furthermore 13 G=9803>11531—10667°—1728? would require 
a somewhat longer operation. 
The factor 11 or 13 can be called the added factor. A danger is 
attached to this method, in the ease of multiplication by an even factor, 
