260 
MU. G. T. BENNETT ON THE RESIDUES OF POAVERS OP NUMBERS 
Let 
determine and suppose 
(lUod 
P' {p^ — 1) ^ = y (mod 2 j). 
Laise bolli sides to tlie power then 
Now 
tlierefore 
and 
therefore 
tlierefore 
therefore 
therefore 
f'yp^ 
/ = 1 + ap (mod ]r), 
f 1 + (mod j/), 
/' = I -h oL p (mod p'), 
1 + OL'y 2 p~^ (mod ]P), 
1 + a'yfp~'= L + ot^/“'(mod y/), 
ya' = 01 (mod Jj), 
y = I' (mod 2 i). 
Now i \ j", /3, yS', and, therefore, f, and finally y, are real numbers, whereas, by 
supposition, k is complex, therefore on this supposition r/ and f are complete 
generators. 
Example .—Modulus 3h 
The highest exponent is 
cB (3^i) = 3^ — 33 = 72. 
3 (3= - 1) = 24. 
We shall find that 2 + 7 and 4 + 37 caii be taken for generators. 
57 = 2 + 7 has exp 24 mod 3'^ 1 
+ = 4 + 37 has exp 3 mod 3^ J 
2 + 7 = (2 + 7j‘® (2 + if (mod 9) 
= (7 + 37) (2 + 77) (mod 9), 
and 
Now 
_/' = 7 + 37 has exp 3. 
/ee I + (1 + 7). 3 (mod 3~) 
/' = 1 + (2 + 7). 3 (mod 3') 
