Art, 1.— T. Takencmchi: 



Uif"^- 1 L 



d^Q (moà. p-l)< {d>p-1 II. 



(/>i m- 



■d = 0(moa.p-l) IV. 



Case I (/=1, d<p-l) 



In this case f(P'0 = P"~'G ,— 1) and the order of 31 isp"" 1 . Let w 

 be an integer which is divisible by p, but not by p 12 ; and consider 

 à numbers 1+ff», a=l, 2, ...., d. Then, since the exponents to 

 which these numbers belong (mod. p") must evidently be powers 

 of p, the classes represented by them are the elements of 31. We 

 shall now shew that these classes form a system of bases for 31. 

 Here, when n^d, the classes represented by the numbers 1 + ^', 

 a=n, n+1, ..., d, all reduce themselves to one and the same class. 

 However, it will be found that the following proof holds even in 

 such cases. 



The proof consists of two parts. First, Ave shall prove that if 

 e a denotes the exponent to which 1 + ^ belongs (mod. p"), then 



fle a =p»-\ (2) 



i,a 



From this we shall see that, by means of the above defined 

 elements, just p 1 " 1 elements are represented in the form such as 

 (1). That these p" -1 elements are all different from one another 

 will be seen, if we next prove that 



Ù (I + *?)*• *\, (mod. p"), (3) 



!, a 

 a / _1 -i 9 j\ excepting the combination e 1 '=e.J = ...=e d —Q- 



To prove (2), let us determine the values of e a . By the 

 binomial theorem, 



1) As for the existence of such integers, see e.g. Weber : Bd. II, 



