4i8 MATHEMATICS: H. S. VANDIVER Proc. N. A. S 
ng(fO ^ n(g(r^^ - - '(P - ^\mod p), {7a) 
s t 
where t = 1,2, , ^ — \ To reduce g{r^^ ~ ^) modulo p, we have 
rri = n + i -j- TTi - r,- + 1 
Raising to the power 2t, and noting that rr,- - r,- + i = 0 (mod ^"), we get 
^'Vf ^ f^V 1 + 2^K- - U + V'l^ (mod 
Transposing, we have 
2«rr+l \rr, - u + i) - r^Vf - rf+ x (mod 
But, by definition, 
+ 1 ^ + ' (mod p^) 
and therefore 
(rr, - n + ^ (rr.. - + O^' + ''<^' " " (mod p'^), 
whence, if ^ = o, 1, , ju- 1. 
2tpJ''-^^q/'^' - ^ r^'Srf - 2rf+ , (mod 
i i i 
Now 
2rf = 2r!'-+ , ^ af + aV + + a'J = S,„ 
where the a's are the integers less than p^ and prime to it. Hence 
ng(r^'-^)=n(£^'(mod/,). (8) 
To reduce the expression on the right-hand side we shall first consider 
the quantity 52^. It will be shown that 
52, = R2t (mod + ') (9) 
for t < (p - l)/2 and 
52, ^ R2t (mod p"") (9a) 
for t = {p - l)/2, where 
R2,= l'' + 2''+ + (^" - lf\ 
We have 
S2, = R2, - (^''' + {2py' + {?pf' + + (?" - (9b) 
= Al" + 2^' + + (p»-i-lf'). 
Now 
T2, = 1'^ + 2^^ + + (^« - 1 - 1)2^ ^ 0 (mod - ^) 
for t < {p- l)/2 and 
T^_i^O(mod^"-'). 
To show this let ^ be a primitive root of p'^ ~ \ then k^^T2t = ^2; (mod 
■y - and 
(^^^ - 1)72,^0 (mod^"- 
In this relation k^^ — 1 is prime to p for / < (^ - 1)/2 and for 2^ = ^ — 1, 
k^^ — 1 is divisible by p but not by Hence 
= 0 (mod - 
