Classes of Congruent Integers. \\ 



constitute d successive integers, no two of them can differ by a 

 multiple of d, consequently the numbers bph + (g b ~jb) d and c+g c d 

 are all different from one another. Therefore, the three kinds of 

 exponents found in (20) are all different. Also they are all less 

 than n, since c b '<.e h , e c '<ce c . Thus, as in the proof of (3), we 

 conclude 



l/(l + ~ , f b ' 0(1 +»«)'•'* 1. (mod. p"). Q.E.D. 



Therefore the numbers (11) represent a system of bases. 

 The invariants are 



■■n-bJa 



■ ,r »-ii> -i 

 h+ L~~d — J, 6 = 1,' 



, \ J h ^L~d J, 6 = 1,2, ■•■ h-\, 6*0, (mod.».) 

 ■when n>k, < * 



I and pir~] c = k,k+\, ■■■k + d-1, ) 



ifn^k + d, (c*0, 



or fl =*,*+i f ...»-i, n moa -*)i 



itn<:k + d, J 



when n ^ k, [ /o ^ir] a= 1 , 2, ■ ■ ■ w - 1 , ) 



* ., . (a*0 



itn>l, > (mo a. p ). 



or a=l, if n=l, J 



Henee the rank is given by 



r=d, if k + dén, ) 



r=n- [— 1 , if \<n<k + d, V (21) 



r = l, if n = \. ) 



It is to be observed that the preceding result (6) is included in the 

 above. For, when d<p-\, k= (r^zyj =1 and > if n<d+l, 

 [— ] =1 ; thus (21) becomes (6). 



§• 4. 

 Case III (f>\, d$0, moä.p-i) 



In this case we shall take as a system of the representative 

 bases the following fd numbers: 



