( 498 ) 
follows: 2362 + 455, 235 « 237 + 456, 234 x 238+-459, ete, then 
the numbers 455, 456, 459, etc. differ 1, 3, 5, etc.; so they form 
a series of order two, of which the (n — 1) term is: 
n° + 455. 
So this is the value of r, whilst (after n operations) the factors 
have become: 236 — n and 236 +n, 
One of these will be a factor of G, = 112303 if 2r + 1 is divi- 
sible by that factor, that is for 
2n? + 911 = p (236 + n), or = p (236 — n), 
The first of these equations gives 
Pre | 
erik Sa eS Dae MeBS, 
The form under the root sign must of necessity be a square and 
p at least 4. 
To investigate which value must be taken for p it may be noticed 
that if p increases with 1, the whole form grows with 2p + 1 + 1888. 
So the algorithm of § I gives: 
p p° + 1888p» — 7288 to which the same abbrevia- 
ee a 441889 £ voce tions as formerly ean be applied 
Piet eaten we it 2177 by paying attention to the 
2x 541889 —= 1899 two last figures of the numbers. 
_ 4076 The smallest factor being 1, 
1901 p can be at most 235 and on 
P account of the combining of 
7 the additional numbers into 
9 groups of 2, 1, 2 and 5 the 
OENE Hasta ning 15601 number of operations is about 92. 
2x 1912 = 3824 The advantage of this method 
(oe ole ae above the direct method of § I 
ESA is, that the numbers are smaller 
951918 | 8836 and the number of additions 
23176 is considerably less. 
51925 = 9625 The number 513667 dealt 
52801 with in § I gives after 3 
ete, 
additions the value of n. 
If in general Ga? + bo, then after n operations we have 
G = (ao + #) (4 — n) + by + n?, so that r —=b 42, The factor ang 
