Classes of Congruent Integers. 25 



instead of (42). For, if (42) has a root x', then (43) must have 

 a root x"=7T-x\ since <7=fr(p — 1). Conversely, suppose that (43) 

 has a root x", and put x"={f}. Then p+{tf {p - l) }=0 (mod. p" +I )> 

 whence follows e(p—})=d i.e. £=&. Hence, if we determine an 

 integer x from x"=it ! x (mod. p' +1 ), then a;' is a root of (42). 

 Thus Ave see that (42) and (43) do or do not admit of solutions at 

 the same time: Q.E.D. 



Also from £(p — 1) = d, it follows that the condition d = (mod. 

 p — 1) need not be stated in the case when (43) has a solution. 



Thus, including all the results obtained concerning the group 

 31, we can state our final result as follows : 



31 is an abelian group whose invariants always consist of powers of p } 

 since the order of the group is p f(r -^. And the rank is given bij 



r=fd+l or fd, if n>d + k, 



according as the congruence p + x p ' 1 = (mod. p tf+1 ) 

 has or has not a solution, 



f=fd, if n=d+Jc, l '''' 



r=jCn- [— 1 ). if l<n<:ä + 7c, 



r=l, if n=l. 



Application 



The problem of determining all the ideals that admit of 

 primitive roots was solved by A. Wiman , and recently by J. 

 Westlund 2) in a somewhat different manner. Here, as an applica- 

 tion of our results, we shall determine all the ideals of the form 

 ttt=p" which admit of primitive roots. 



It was shewn in §. 1. that if p f —l=p l "ip 2 "-2--- jV" 1 , where p u p it 

 •••, p„ are distinct primes, the invariants of 35 are Pi"\ ;V' 2 , " , p"' 1 - 



1) Ofversigt af Svenska Vetenskapsakadeu.iens Förhandlingar, 56. 



2) Mathematische Annalcn, Bd. 71. 



