Classes of Congruent Integers. 17 



accordingly in choosing f, we need only avoid the ]/"' values 

 which are congruent to 



>{b> £ ) + (i°^ 



and this is always possible, since N(p)=p f >p f -\ That £„ thus 

 determined is not divisible by p can be seen at once by putting 

 c„=T, c 3 =c s =--=c / =0 in (33). Hence if we consider a number 

 1 +fX f+t , the exponent to which this number belongs is 



This premised, let us consider the product 



p = (l + çy+*f' d +\ ii(i + Çjz ,: / H , 



2,/ 



e' M =0, 1,2, ...,e„-l, (i = 2,3, •■•/), 

 e'„ + * = 0,l,2, ..., ClJ+t -l. 



Then, just as in (29), when at least one of the exponents e' u is not 

 divisible by p, 



P=l + {v"}. (34) 



When all the exponents c' ki are divisible by p, we put 



e f H=p l+ °. c lt e' d+k =pi. c a , 



where p"(g—Q) is the lowest of all the powers of p contained in 



- LJ2l - and e'rf+ij then 



p ' 



p=l + L(E c ß i ) + (2 c ß.y K d +c ßAp K K + , 



l s,r 2./ J 



hence by (33), P=l + {f Hl+s)i }. (35) 



Thus, here (34) and (35) playing a part of (20), we can shew 

 exactly as in §. 3, that in the form 



_/i +f -i«A t %+^ e',-0,1,2,...,^-!, 



a=l,2, ■••, d + Ä— 1, « ^ (mod. j)), 

 •=1,2,.»,/, 



