18 Art. I.— T. Takenonchi : 



are represented e d+k ™=A T numbers which are all incoiigruent 



(mod. p"). We may notice that <?,, +fc =p L d J =p l d = — , and 



also that e M =e k =pi- d J if i=£l. For, since by (32), x=? { (i=f=l) can- 

 not be a solution of (31), we have 



(l+?y) p =l + {p* +d }, ifi+1; 

 whence follows e ki =j> d ( l= £l)- Therefore 



]y = _e L /^ /(n-l)-l 



Hence, if we consider the product 



B=(l+|^) e ' H . Q, e' H =0, 1, 2, - « n -l, 



then e /d . p^"- 1 '- 1 numbers will be represented by it. However they 

 may or may not all be incoiigruent. At any rate it is certain 

 that these numbers represent all or a part of the elements of 21 

 repeated the same number of times, say h times. To determine 

 h, let us consider the number of ways in which R = l (mod. p") is 

 satisfied. In the first place it is necessary that e' kl = (mod. p); 

 otherwise, we shall get /7 (1 + ?^) M =l+{p*}, consequently R$ 1 



(mod. p"). Next, for each value of e' M the accompanying factor Q 



e' 

 must be uniquely determinate; for if we suppose (1-f ?i^ 1 ") kl . Qi=l 



and also (I+^jt*) 6 * l . Q» = l, it follows Q y = Q^ which is im- 

 possible as shewn above, unless Qi=Q 2 . Therefore h cannot be 

 greater than the number of the multiples of v contained in 

 0, 1, 2, ••• e kl — l, i.e. hé-^-. But, on the other hand, since R re- 

 presents e ki p fin ~ n ~ l (^j/^ -0 ) elements, while there are only pft* -I) 

 elements in 31, we must have h^— — a ' n H = — . Hence we 

 obtain 



h=-^- ana en-pto-n-isaW». 



