228 
MR. G. T. BENNETT ON THE RESIDUES OF POWERS OP NUMBERS 
Tlien we must have 
aQ= 1 (mod 2'') 
with exp a power of j) (or unity) mod 
„ p ( ) mod Po" 
0 
occ. 
Let 
be a number with exp mod P;^^^ 
y -2 „ „ mod Po^ 
&c., 
(if in any case ]p is not a factor of (P'^) y is = 1). 
Then we may put 
= yL (mod P^^^) 
tto = yP (mod Po^-), 
&c., 
and therefore 
a = ^0 + + yHz + • • • (mod m) 
a = (4 + + • • •) (^0 + + • • •)'' (^0 + + 72^2 + • • •)'' • • • (mod m) 
= 9\^9i' • • • (mod m). 
The first factor and each of the following in which the y is = 1 is = 1 (mod m). 
The number of factors remaining is the number of the principal factors P^ which 
have a power of y) as a factor of ^ (P^). 
These factors generate the ^-power-exponent numbers : corresponding to each set 
of indices i is one of the numbers a and vice versd. 
These generators are only a very special kind of j:)-power-exponent generators 
inasmuch as each is congruent to unity for each hut one of the principal factors of m 
as modulus. They will be called unitary generators and will be useful for the 
discussion of the more general type. 
In the case when ]:> — 2, let 
f 1)6 = 2'" — 1 or 2''“^ — 1 (mod 2'") 
yQ have exp 2'^“^ (mod 2") 
(if K — 2, f does not occur ; if /c = 0 or 1, neither g nor /’), 
y^ have exp 2''i (mod P/^) 
73 2"^ (mod Py^), 
&c., 
