Classes of Congruent Integers. 23 



ff(i+e^ rt =i+{p+(AfO^" (, "Y}(i<vfi) p, *-p'» r *+ 



here #i> or =0, according as the congruence 



p+(ie&) P " ( *- 1> . ^ = (mod. p" +1 ) 



does or does not hold. If we determine ," from p = ~ d p (mod. p' ,+1 ), 

 then this congruence can be replaced by 



p+(ic t £.)P 3 V"=:Ö (mod. p) 



th at is p + (i ejf^V" 's0 ( mod. p). 



Therefore, if the congruence 



l>-\-x"- l = (mol p) (41) 



has no solution, then «1=0 and it follows as before that the num- 

 bers (28) represent a system of bases, consequently r=fd. 



If, on the contrary, (41) has a solution x=x , then other 

 solutions are x=2x , 3x , ■■■, (p~l) x , and they are exhaustive. 



Now, from (23) we get ~(cßf )^0 (mod. p); whence follows 

 that we may put 



x = x = S a c f f ", 



where a t (i=1, 2, ••■,/) are rational integers and we may suppose 

 «,^0 (mod. p) without losing generality. Then we can determine 

 an integer ?„ so that 



pfl cßt) + (i cßfj + cß * (mod. p), 



c , c 4 = 0, 1,2, -.p-l, (mod. p), (i=2,3, •••,/) 



excluding the combination c = c 2 = c 3 = • • ■ = c } = (mod. ^>). 



Tli en in the form 



Q = (l + f :*»/** & kl + tef" , e ai =0,l,2,-,e ai -l, 



(l+?F l ) e ' al *'*n=0,l,2,...,e -+t -l, 



