785 
Mathematics. — “The primitive divisor of am—1. By Prof. J. 
C. KLuyver. 
(Communicated in the meeting of November 25, 1916). 
Tbe binomial equation «”—1=0 has M=y(m) special roots, 
which do not belong to any binomial equation of lower degree. 
Denoting by r the integers less than m and prime to m these special 
niv 
roots are of the form #,=e * and the product 
Fte) = (te) 
is called by Kronecker the primitive divisor of «”—1. 
It is shewn that Z, (e) cannot be resolved into rational factors 
and that the decomposition of w”— 1 into rational prime factors is 
given by the equation 
om — Ls Fg (a), 
; din 
where d is successively equal to the different divisors of 7, unity 
and mw itself included. 
By inverting this formula in the usual manner, we infer that 
=) 
De (2) -—— vill Nis ( ) — II (ed— | ytd _ (1 Bo wdyu(d’), 
is dh d/ in 
(dd = m) 
In this fundamental equation «(d') stands for zero, if d' has a 
square divisor and otherwise u(d') equals + 1 or —1, according 
as d' is a product of an even or of an odd number of prime numbers. 
From this expression of /, («) the following properties of the 
primitive divisor may be deduced. 
LIF on =n,n, and n, and n, are relatively prime, then 
»(2) 
Pel MEE (20) a]. 
has ari 
Il. The greatest common measure of #,, (a) and A5, (a) is Hin, (+), 
n, and n, being prime to each other. 
IL. If m has at least two different prime divisors, then A (1) =1, 
but Bn (1. p. when m isa prime number p or equal to a power of p. 
IV. If m resolved into prime factors is of the form m= 
denotes the produet p,p,.-. pr then 
0 
Rapp and m 
m 
F nl) —- dE 0 (ammo) . 
From this proposition it follows that in order to find a definite 
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