( 433 ) 
Example: G=51=8X19. The factor 6 would give: 
37\2 1\2 
G, = dag = 18 19 = (=) — (5). 
! x 2 2 
To avoid fractions we can in such a case moreover multiply by 
4 and so write 
Gy = (2 X 18) X (2 X 19) = 372 — 12, 
Example: G = 100895598169 (see introduction) if we know, that 
the factors contain an equal number of figures. 
The number 1008 formed by the 4 first figures of @ can have 
oriomated trom Lis 99 tae 84. ES KOE oS Lo ol 
The greatest factor is thus at most 9 times the other. So we 
might try whether 3G, 5G, 7G@ or 9 G after extraction of the 
root would immediately give the difference of two squares. If this 
is not the case, we might do the same for 
4x26, 4x 4G, 4x6G and 4x 846. 
The number 4 Xx 8 G immediately gives a result U). 
XI. Application of the results of the theory of numbers. 
A few properties out of the theory of numbers will be discussed 
here briefly : 
1. Every odd number G whose factors are 4-folds +1 can be 
decomposed into the sum of two squares in 2"— ways, if n represents 
the number of different prime factors. 
For this the squares are supposed to be respectively indivisible. 
Example: 325 = 5?13 = 12+ 18?= 6217? (but the decomposition 
10°+-15? is to be left out of consideration). 
The above mentioned decomposition which is rather prolix, if accom- 
plished by the operations of the theory of numbers, becomes pretty 
simple when the reverse of the method of § I is applied to it. 
Example: G= 953 = 30° + 53 or 
BESS bP ESE. il. eet 9 KDE 
1) See the note in the introduction. 
