J(3 Art. 1. — T. Takonoucbi : 



numbers (28) represent a system of bases, and consequently r=j'd. 



If (31) bas a solution, say x=x , tben since evidently .r o ^0, 

 the/) — 1 numbers x , 2x , ■ ••■( J p — l)a; are all incongruent (mod. p) 

 and all satisfy (31). Thus we get p— 1 solutions; but no more, 

 for (31) is of the (p — l)</i degree with a prime ideal modulus. 



Now, it follows from (23) that the // numbers 



cß].+cß 3 + +c,f/, Gi=0, 1, 2, ■••p— 1, (mod. p) 



(*=1. 2, ■■■/), 



constitute a complete system of the representative classes with 

 respect to mod. p. Hence / rational integers u„ i=\, 2, •••,/, 

 such that 



x = aß J + a^f 3 + + aß }, (mod . p) 



can be uniquely determined with respect to mod. p. And since 

 x ^0 (mod. p), at least one of these ffl's is not divisible by p. 

 Without losing generality, we may suppose (', + (mod. p). 

 Then, if we put the other p— 2 solutions, 2x 0J , (p— l)ar , in similar 

 forms, the coefficients of $„, being congruent to 2a u 3«,,---, (p— 1)«, 

 respectively, are all not divisible by p. It follows therefore that 



the numbers of the form cß 3 +cß 3 + + c / ? / can never satisfy (31). 



Thus 



p + flcß^P" 1 *0, (mod. p). (32) 



Hence, we can determine an integer ç so that 



^(Ecß^ + flcß^ + cß^O, (mod. p), (33) 



c , c t =0, 1,2, -p-l, (mod. p) 

 (i=2, 3, .../), 



excluding the combination c = c* = c 3 = •■■ c f =0, (mod. p). 



For, when c = 0, (33) reduces to (32) which is always satisfied ; 

 when f„#MJ, if Ave determine a rational integer c from cc = — 1 

 (mod. p), and then (33) becomes 



e, * c,o fie, + c (ic, f .V s iffice A) + (poeß^fl 



