( 435 ) 
The number of factors, 4-folds —1, must necessarily be even, 
as only an even number of 4-folds — 1 can produce a product 
4-fold + 1. 
4. If a A-fold + 1 can be decomposed only in one way it is not 
necessarily indivisible, but i¢ may contain factors which are 4-folds —1; 
the number of those factors amounts to two less than the number 
of factors which are 4-folds + 1. 
Of the above-mentioned properties we can make use in the 
following way: 
Example: G = 898423, 4-fold — 1, so it is not to be decomposed. 
In the first place let us determine the greatest possible number 
of factors in the following manner: 
The product 3 X 7x11 Xx 13 Xx 17 X 19 = 969969, thus more 
than G, so that the number of different factors of G cannot amount 
to more than 6. 
If we leave out 3 and 7, it is evident that 11 « 13 x 17 x 19 x 23 
is also > G. Likewise 13 xX 17 x 19 x 23 x 5 is > G, so that 
G can contain at most 4 different factors. These might be: one 4- 
fold — 1 and three 4-folds + 1, or three 4-folds — 1 and one 4-fold + 1. 
By leaving out 13, 17, 19 and 23 we find 293137 ae 
3) 
so that G can have at most three different factors, namely one 4- 
fold — 1 and two 4-folds + 1, or three 4-folds —1. 
We have supposed here that the numbers left out are not divisors 
of G. 
We now multiply G by 3 to obtain a 4-fold + 1, and by two 
prime numbers more, 4-folds + 1, non-divisors of G, for instance 
5 and 13, and we test whether 13 x 5 X 3G can be decomposed. 
If this should prove possible G would really possess three factors: one 
4-fold —1 and two 4-folds + 1, which number has been increased 
by the added factors to two 4-folds — 1 and four 4-folds + 1. 
If 13 X 5x 3G cannot be decomposed, there can still be three 
factors 4-folds — 1, and we should have to try 17 Xx 13 x 5 & 3G. 
If this can be decomposed, G is indivisible, for only in that case 
the number of factors 4-folds + 1 (namely 17, 13 and 5) is two 
more than the factors 4-folds — 1 (namely 3 and @). 
The last number to be calculated being rather great, we can follow 
another method by remarking that PG is nearly 96; so G is smaller 
than 97 X 101 X a number smaller than 103; so that if we have 
