ON THE THEORY OF NUMBERS. 137 



...\l<(fi K ~~ 2 ), which is a symmetric function of /3, /3 3 , . . . j3 x ~ 2 , the roots of 

 the equation 2^ + 1=0, is therefore congruous to *//(y) ^(y 3 ) •• • • *Kr x_2 )> 

 which is the same function of y, y\ y ', . . . y A ~ 2 , the roots of the congruence 

 ^-1-1=0, mod X. Hence the necessary and sufficient condition for the 



P s 



divisibility of — s -_- by \ is that one of the fi congruences included in the 



formula 



^(y 3 " -1 ) =0, modX,M=l, 2, 3. ../i, (a) 



should be satisfied. Now y^ 2 "" 1 ) ^(y 2 ' 1 " 1 ) =6 yf _1 4-^yo 2 "" 1 + W'" 1 



A — 2 



+ ... +K-2y 2n ^ll> or > observing that y , y^ y 2 ...yx_ 2 are the numbers 

 1, 2, 3, ...X — 1, taken in a certain order, and introducing the values of 



* > Oj> "2> * ' ' 



y -(2»-i)^( y 2»-i) == s a- 2 "- 1 I H, mod X. 

 x=l X 



This last expression may be further transformed as follows. lf/(x) denote 



x=x 

 any function of x, and F(#) = 2 /(a-), we have the identical equation 



x=l 

 x=\ — l „ .r=y — 1 / > \ 



2 l2£./(:r)+ S F(l^Wy-l)F(X-l), 



y and X being any two numbers prime to one another. To verify 

 this equation, we may construct a system of unit points in a plane ; 

 then the right-hand member is the sum of the values of f (x) for all unit 

 points in the interior of the parallelogram (0, 0), (X, 0), (X, y), (0, y) ; 

 while the two terms of the left-hand member represent similar sums for 

 the two triangles into which the parallelogram is divided by its diagonal 



yx—\y=0. Writing then in this identity a; 2 " -1 for f(x), and employing 



x=x 

 the symbol F 2B _, (x) to represent the sum 2 a 2 " -1 , or rather the function 



x—\ 



g^ + jaaw-i + B, n ' gn ~ 1 ,sii-s_B, — n - 2n ~ ] — x*«-*+.... 



2» 1 n.2w-2.n.2. -II. 2»— 4.11.4 



V y " _1 n.2.II.2w-2 



in which B x B 2 ...B re are the fractions of Bernoulli, and which, when x is 

 an integral number, coincides with that sum, v\e find 



*=X— 1 „ x=y — 1 r > "] 



a;=l x b=1 L y J 



But F 2 „_! (X — l)=F„ 2 _i (X)-X 2 "- 1 is evidently divisible by X; so that 



x=x — 1 v r *=y— I r x --i 



2 * 2 »-i I^+ 2 F 2n _! I— =0, mod X. 



a=l x x=l L 7J 



The congruences (a) may therefore be replaced by the congruences 

 *=y— 1 r x -, 



2 F 2n _i i_ = 0, mod X, which may be written in the simpler form 



x=l L 7 J 



