( 335 ) 
unt'l a square is obtained. As 83 is a primenumber, one will be 
foreed to eontinue until 83 has been added; so the number of 
83—21 . 
-—- 1= 32. Now if in Table II *we 
additions amounts to 
proceed horizontally from the base 83, this number is not found in 
the oblique line 4(a—l); so we can pass to the line 3(2a—3). This 
means however that it is not necessary to continue the additional 
A rod IN? 2 
operations until (=) = 42°, but only until (es = 165 
EP ) 
2 
the base of which is 31, as this number is the base of the larger 
one of the two numbers 81 and 87 between which 83 is situated. 
If it is found that 83 is not a threefold, we can pass from the 
line 3 (2a—%) over the lines 8 (a—2), 5 (2a—b) and 12 (a—3), 
to the line 7 (2a—7). So the operation can be stopped at 
In the case of the number N= 112303 (see the introduction) 
one would have to perform nearly 398 additions, (if the numbers 
to be added were combined in groups of 4 and 6). If however 
division proves that none of the numbers 3, 7, 11, 13, 17, 19, 23 
is a factor, only 211 additions are necessary. 
Ill. Determination of non-divisors. 
If we put N=ab+c, any codivisor of a and ec or of b and 
c will be divisor of N, whilst a divisor of ¢ relative prime to a 
and 6 cannot be a divisor of N. 
Example NV = 73489207 !). 
We put NV = 8573? — 7122 
or = 8573°—12—7121 = 8574 6572—7121 
— 23142942143 —T121: 
this proves that 3, 1429, 2143 and 7121 are non-divisors of N. 
1) This number was not obtained by multiplication of smaller numbers but chosen 
arbitrarily. Likewise all the other numbers of five and more figures decomposed in 
this study were chosen at random, the number mentioned in the introduction excepted. 
