Classes of Congruent Integers. \ 3 



contained in <.'„, (supposing <.•',„ =£0), and put e' ai =p 9 ' c ai . Then ex- 

 panding (l+^) e «i and taking notice only of such terms that 

 contain p in the lowest power, Ave get 



( 1 + f.a« , e « = ( 1 + Ç o 7T ')' W ^ =(1 + C..Ç.JT •+ ■ ■ T' 



= 1 + 0..?;^ ''"' + , if d-cp— 1, 



or d>p— 1, isaj-, 



= l + (c, ! i ; y'"-" ; ' : '' + if d>p-\, a<A; g'<.j a , 



= 1 + (cjfy - j "-"^'" + , if d>p - ] , a<Ä-, j7'S /.. 



Therefore, if we denote by /'" the lowest of all the powers of p 

 contained in the / numbers c '<,,-, i=l, 2, •■•,/. and put e' ai =p 9 c ,-, 

 (consequently at least one of c ai is not divisible by />,) then 

 II (l+Gvf ai is equal to 



or l + î(c a .J i f.^'P' + , 



or 1 + È(c ai ç ,./"•/'" J "-"^" + • 



Now, by (23), -(^.f,)*° (mod. p), hence also ^(c^^^O (mod. p), 

 m being any positive integer. !) Therefore 



or l + {p^'}, V (26) 



or l+{p^'' 1 - f(ï --V ; }. ) 



Thus we find that the product on the left hand side of (25) 

 consists of d factors, each of which having one of the forms (26). 

 If we compare (20) with (20), evidently the further steps to be 

 taken here are just the same as in the proof of (15). 



Let us therefore go straight to the conclusion. 



The fd numbers (22) represent a system of bases. If the 

 invariants be wanted, we have only to write down the invariants 

 given in §. 2 or §. 3 (according as d*<p— 1 or d>p— 1, d^Q, 



1) Weber: Bd. II, § 167. 



