ON THE THEORY OF NUMBERS. 975 
Wy, StY 
1,01,” 8 
because My, Tm,“ =(—1)'t™°F), mod p, by Sir J. Wilson’s theorem, 
while My,“*=1, since p,¢+=0: whence, multiplying and writing s, for 
(<1) = 
p,©, we obtain the congruence written above. Let r represent any term of a 
system of residues prime to p—1; let the numbers m,,m,'...m;‘” be deter- 
mined by the congruences m, =m,© r, mod (p—1), combined with the 
condition 0 < m,? < p—1; and let m+m'+... +m =n, (p—1)+5,, 
where again 0 <s,<p—I1: we have for every value ofr an equation of the 
form a , (0) 
x (G")=p"* kK —— , (0) and a congruence of the form 
®, (y) each o7—N,, IIs, 
v (y) =) Imm’ Tm," ... 
- Let y(0)=A,+A,0+...+Az 6", k+1 denoting the number of terms 
in a system of residues prime to p—1; let n, be the least of the numbers 
re Ny +p and j the exponent of the highest power of p dividing 
A, ie. ..A,: then shall j=n,. For, first, if 7>n,, from the equation 
, mod p. 
yy, On x ic =®@, (9), in which the coefficients of the powers of 6 are integral 
sll we infer the congruence W, (vy) * LY) =, (y),mod p; but x Ca X) 0, 
mod P3 therefore, ®, (y) == 0, mod p, which is impossible, Eanes if 
j <1, writing A; for A;+p/, and observing that W, (y) is prime to p, for 
every value of 7, we find ‘ 
Ay +A,’ y PLA yrt. Be 8). Gi ps5 mod pv’ 5 
for every value of r: but the sees iat of this system is prime top, there- 
fore A’, =0, A,’ =0, A',=0, mod p,-/, which is contrary to the eka 
thesis that j j <7, and that p) is the highest power of p dividing A,, A,, 
Ax 
The application of these results leads to the following general theorems ; 
in the enunciations of which p is an uneven prime, and A a number not 
divisible by any square. 
“Tf A=4m+3, p=An-+1, and if we represent by a and 6 numbers less 
than A and prime to A, respectively satisfying the equations 
b 
(5)=1, G)= —1, we have 
- 3b— sa 
4p 
=2°+4 Ay’, 
>) 
CXS) ace oS 
f1, (Man) 
“Tf p=4An-+1, A being of any other linear form than 4m-+3, and if we 
represent by a and 6 numbers less than 4A and prime to 4A, respectively 
satisfying the equations (== 7 *)= +1, (5 ; =*)= —1, we haye- 
