Mathematics. — *'The primitive Divisor of x^ — 1." By J. G. 

 VAN DER CoRPüT. (Communicated by Prof. J. 0. Kluyver). 



(Communicated in the meeting of September 29, 1917). 



This paper is an extension of tlie article of Prof. J. C. Kluyver : 

 "The primitive Divisor of a:"^ — 1." (These Proceedings, Vol. XIX, 

 page 785. 



27111 



Definition 1. If k be a positive integer, the product n [x — e^ ), 



extended over all the values o- of a reduced rest-system, modulo k, 

 is called the primitive divisor F]c{.v) of .t-^" — 1. 



Definition 2. If k be a positive integer, 'f = (f{k) represents the 

 number of positive integers ^ /■, which are prime to k. 



Proposition 1. If k be a positive integer, then the primitive 

 divisor of .r^ — 1 is a polynome of the degree (f\ 



Definition 3. Tlie numbers A,, {fp ^ }. "^ 0) are defined by the 

 relation 



9 



Fk (^) = ^ Mx', 



/ = 



k being an arbitrary positive integer. 



Definition 4. In the functions /> {n,k) of the variable integer n 

 {k l)eing a positive integer), which are called the arithmetical chai-acters 

 of n, modulo k, r represents an arbitrary integer, prime to k. 



The functions /«(n,/) and y^A'ijk) are identical or different, according 

 as II and v are mutually congruent or incongruent, modulo k. 

 Hence it follows, that there are (p different arithmetical characters 

 Xv(r?,/(:), modulo k and these functions possess the foil > wing properties: 



I. 7.-,,'m,k) XAn,k) = y^{t}in,k). 



II. y^./jii ,k) = y;{)i,k) if m^n {mod. k). 



III. The modulus of x-,{n,k) is equal to or 1, according as ?i 

 and k are commensurable or incommensurable. 



IV. x-^{>i,k) is equal to or 1, according as n and k are com- 

 mensurable or incommensurable. 



V. x-iC'^'^') 's equal to the symbol ( -7- ) of Legendre. 



VI. xv(l.^) = 1. 



