Classes of Congruent Integers. 



r=d, if d<:n, \ 



r=n-l, if l<znéd, I (6) 



r=l, if « = 1. J 



§• 3. 

 Case II (/*=1, d>p— 1, tZ^O, mod. p— 1) 



As in the last paragraph, we have (l + s"J"=l+ {p a+ ' i } + (p"*}. 

 But since fc>p— 1, d^O (mod. p— 1), we get two cases « + ti^ap, 



according as d<^a(p—l). If we put _ , I =k (fc>l), then 



d + ;r') p =l+{p' + <'}, when a^k, 

 = l+{p"*}, when (KÀ-. 



Therefore, if a^Jc, we get successively 



(1+^=1+ {p ,+ " d }. ) 



(l+»y='l + {p ,+M }. > (7) 



But, if a<fc, we get 



(I+-y'=(l + {p 1 >})< J =l+{p"*+"}, when aj>Sfc, 

 = 1 + {v' vp2 } , when o^<&. 

 Proceeding in this way, we conclude that, if a<h, 



(i+*r=i+.{rt. (8) 



i=0, 1,2, ••/„, 

 where 7a is an integer such that flp Jtt <kéap :>a , or, in other words, 

 /«= r%= —J ; and further 



(l+^ +1 =l + {^^}, \ 

 (l+^f Ja+2 =l+{p°>+ M }, > (9) 



From (7), (8), (9), we get 



