258 MR. G. T. BERRETT ON THE RESIDUES OF POWERS OF NUMBERS 
it follows, since cC — h is a pure number, that 
a’’ — 6 = 0 [mod (a — /3i)^], 
therefore 
rP —6 = 0 [mod 
cC = 6 [mod + y8'“)^]. 
Hence for the modulus (a + ^0 ^ the residue of any power of a number is the same 
as for the modulus 
Now is a real 
Therefore the exponent to which a belongs for modulus (a -|- is 
^ if X = s, 
t (a^ + ^~Y~^ if X > 5 , 
where t is the exponent of a for mod or -j- /3^, and [a? -j- is the highest power of 
otr + that divides a* — 1. 
The greatest exponent is 
and 
Cl. (a + = (a- + - (a^ + + /3’ - 1) («' + 
Thus for powers of a mixed prime primitive roots do exist. 
(xiia.) We see from the last proposition that for a powder of a mixed prime as 
modulus primitive roots exist, and any one of them generates by its powers all the 
residues prime to the modulus. But in the case of a power of a pure prime p)^> 
highest exponent isp^~^ [p^ — 1), whereas <3> [p^ = — [). 
We wish now to find how all the — 1) numbers can be generated. 
lake a number (j with exp p^ ^ (p“ 
fiid„ „ / ,, ,, 
(J can be expressed as a j.roduct 
Avheie 
(J =fh (mod pY, 
f has exp p^ ^ 
h 
5 ) ) > 
p 
:-i 
•, Proposition x. 
/‘and f are each = 1 (mod p) 
>, Proposition xn. 
1 (mod ^^2) I ^ 
