Classes of Congruent Integers. 21 



numbers not only of the form l + {p d ^'} but also of the form 

 l + {p' i+i+:r -'}, .r 2 >0. Hence the number of the numbers of the form 

 1 + [p' m ] which can be repesented by the remaining fd numbers 

 is certainly less than <p (p""''"'"'.) ; whence follows that some elements 

 are not represented. 



Next, let us try to reject, if there be any, a number 1+?^* 

 which belongs to the exponent h. Here, of course ß>k, and for 

 convenience we suppose i=l; similar reasoning applies to other 

 values of i. Now let us consider how many numbers of the form 

 l+{p Ä ] can be represented. Here we must have 



e ai = (rood. p 3a+l ), if a<k, ap 0a <:ß, . 



e ai = (mod. p'"), if a<Zr, ap 3a ^>ß, f 



e ai =0 (mod.jj), if ke.a<ß, I 



(i=l, 2, •••,/). ' 

 Further e ai (a>ß, i= 1,2, ■■-,/), e ßi (i— 2, 3, ■••,/) and e' d+k may have 



any values, provided e ßi ^0 (mod. p) at least for one value of i. 

 The number of such combinations is 



a<4 a«c 



{ n ii^l n n^){ n n^un^n^-n-f.)^ 



Observing that the number of values of a in the first and the third 

 brackets are together ß—k, and also that e d+k =e ei by supposition, 

 we can transform this product as follows: 



o<fc a<k 



1 ///7 g./, 1 \ 

 "V'-*>" ^"i^V ^"'J (39) 



=F«-«(l-^) 



