12 Art. 1.— T. TakenoucM : 



l + £,7r\ o = l,2 f 3, ....Ä+Ä-l, ^=r— Zt] - 1 ') 



a^O (moà.p), > (22) 



•=1,2, ...,/, ) 



where ? 's are integers such that 



ci? i + c.ß, + + Cf £>* (inod. p), (23) 



for all combinations of c's, c ê =0, 1,2, • •■ p— 1 (mod. £>), 

 (*=1, 2, ■■•/.) 

 excluding (^sCjS-'-StysO (mod. p). 



That there always exists such a system of ? 's can at once be seen ; 

 we may take, e.g. the / numbers 1, to, o?, •■•ft/ -1 , where w is a 

 primitive root of p. u It is evident that none of them is divisible 



by p; for if ^=0 (mod. p), then putting d=c 2 = =c i _ 1 =c i+x = 



=c / =0, Ci =l, we get c? 1 +c^ 2 +---c l ? i +--- + c^=? i =0(mod. p), which 

 is contradictory. 



Now, to shew that the numbers (22) represent a system of 

 bases, we must prove the following two relations: — 



j'lile ai =p^- l \ (24) 



i./ 



where e ai denotes the exponent to which 1+f^' belongs (mod. p"). 

 And 



Z7.fl r (l+f l jf t ) 6 «'*l (mod. p"), (25) 



e' ai =0,\,2,---e ai -l, 

 excluding the combination where e' ai =0 for all values of a and i. 



Since fj+0 (mod. p), the number 1 + ?^ 1 evidently belongs to 

 the same exponent as 1+t\ Hence Cai=c a2 ='--=c ( , / =:e a) and in 

 virtue of (2) or (14) 



II ile ai =fle f a =p^- l \ 



Thus (24) is proved. 



Next, to prove (25), let p 9 ' denote the highest power of p 



1) Weber: Bd. II, §. 176. 



