FOR ARY COMPOSITE MODULUS, REAL OR COMPLEX. 
207 
When the modulus is a power of an odd prime. 
(i.) Primitive roots always exist. 
(ii.) The determination of primitive roots depends on a knowledge of those of 
the prime in question. 
Examples .—For modulus ’2® = 32 the numbers whicli belong- to the different 
exponents are 
Exp 1, 1. 
Exp 2. 15. 17. 31. 
Exp 4. 7.23. 9.25.(± 1 + 8 (mod 16)). 
Exp 8. 3. 11. 19. 27. 5. 13. 21. 29 (db 1 + 4 (mod 8)). 
The residues of powers of 3 are 
3. 9. 27. 17. 19. 25. 11. 1. 
Multiplying each by 2^ — 1 = 31 we get 
29. 23. 5. 15. 13. 7. 21. 31, 
and, multiplying by 2^ — 1 = 15, we get 
the same set. 
13. 7. 21. 31. 29. 23. 5. 15, 
The Table of Indices for Generators 3 and 15 is— 
0 
1 
1 
2 3 
4 
5 
G 
7 
0 
1 
3 
1 
9 27 
17 
19 
25 
11 
1 , 
)5 
13 
7 -21 
31 
29 
23 
5 
(Index 
of 15.) 
(Index 
of 3 .) 
(17). From propositions (9), (12) and (13) the exponent to which any number a 
belongs for modulus m = ... is readily determined. 
For by (9) the exponent is the L.C.M. of the separate exponents of a for moduli 
2 q . separately. 
These exponents are separately determined by propositions (12) and (13). 
The greatest possible separate exponents are 
2 ''-" if /c5 3 
2 
1 
if /c = 2 
if K = 1 
>-for mod 2^ (p — 
for modp^, &c., 
and, hence, the greatest exponent possible for modulus m is the Ij.C.M. of these. 
