Vol. 6, 1920 MATHEMATICS: H. S. VANDIVER ,417 
^" ~ ^a, where a is prime to p. Hence r'^ = ( — 1 + ~ ^a)^ = — 1 + 
p^a + p^" ~ ^ai, where ai is a multiple a2a of a. Thus 
2m- 
whence 
and since 
we have 
s pa 
(1 + a2 - ') i_ ^ ng(0 
= Ug{d') (mod (3) 
s 
Now Tlg{d^) = a, where a is a rational integer. Consider the expression 
s 
Ilg{x^) — a, where x is an indeterminate. This is a polynomial in x which 
vanishes for x = 6, and hence is divisible by V{x) = Il{x — 6^), i ranging 
over the (p{ix) integers less than and prime to /x, since V{x) is irreducible 
in the domain of rational integers. We write then 
Jlg{x') = a + V{x)W{x), (4) 
where W{x) is a rational integral function of x. Let 
Vs = v,{x) = n(^ - coO 
i 
where i ranges over all the integers less than 5 and prime to it, and co is 
a primitive s^^ root of unity. Then 
^\ = v,v,y,, , (5) 
X*^ — 1 
where Ci,C2 are all the numbers of the form p^ ~ ^k, where ^ is a di- 
visor of ^ — 1 which is<^ — 1. Since r is a primitive roofc of p^ it is 
also a primitive root of p. Also, (5) gives 
■= V,{r)V„{r)V,Xr) (6) 
— 1 
Now no Vc can be divisible by p since it \vould then follow that 
r' = 1 (mod p) 
which is impossible since c is not a multiple of p — 1. Since the left- 
hand member of (6) is divisible by p, we ,therefore, conclude that ^^(r) = 
0 (mod p). Substituting ^ = r in (4) and using (3), we have 
ki = Ug{r') (mod p), (7) 
s 
By Fermat's theorem we have 
