( 327 ) 
In the preceding century the theory of numbers was built up by 
Gauss, LEGENDRE, LEJEUNE-DiricuLer, RIEMANN, TSCHEBICHEFF 
and others who obtained very important results. But the value of 
these results with reference to the factorisation of large numbers 
can be derived sufficiently from the following two citations: .. . “dans 
“létat actuel de la théorie des nombres on ne connaît aucun procédé 
“direct pour la recherche des diviseurs des nombres ayant plus de 
“dix chiffres dans le système décimal” (Enovarp Lucas, Théorie 
des Nombres, Tome premier, p. 333) and “Les méthodes de Gauss 
“seraient impuissantes à résoudre le problème proposé par MERSENNE 
à Fermar’ (Epovarp Lucas, Récréations Mathématigues, Tome II, 
p. 231). 
Any number MN can be factorised, if it can be thrown into the 
form a°—b® under the condition a >b +1. For then we have 
N= (a-+ b)(@ —b), which proves that the decomposition can be 
performed by seeking a square 0? that, added to N, furnishes a 
new square a, This simple property is also mentioned by FERMAT, 
but apparently not made use of to factorise large numbers either 
by him or by another. 
By means of a table of squares the factorisation of any number 
N can be performed. 
However it is more convenient to determine the square root of 
N and to increase the last figure of this root by unity. 
Example V = 1073. 
/10 | 73 = 38, so 10738—33?— 16—332— 42-37 29, 
yee 
173 As a rule the required-result will not 
63.3 = 189 be obtained so soon; rather we shall find 
= 16 in general N= aj? — bj. 
By adding » to a, we find 
N= (a, +n)? — (bj + 24, n+ n°); 
so we can reach our aim by choosing ~ in such a manner that 
bj + 2ajn 4 n° is a square. 
a number of twelve figures, wherefore did he not apply his method to this number 
of ten figures? May we not conclude from this that FerMar was in possession of a 
special method for special numbers, and that he had dictated to MERSENNE a condi- 
tion to which the proposed number had to satisfy? The correspondence between F. 
and M. may enlighten this point. In that case also could be decided if any of the 
methods given here be related to the method of Fexmar, 
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