256 
MR. G. T. BENRETT ON THE RESIDUES OF POWERS OF NIDIBERS 
The number of residues altogether (prime to the modulus 2 ^) is ‘I’ (p)- Hence, in no 
case can there fail to be cf) (t) numbers with exponent t, t being any divisor of (p). 
Corollary .—In particular, any prime modulus has numbers with exponents <1^ (p), 
i.e., has primitive roots. 
The number of these primitive roots is (/> [<h (p)]. 
If p is a pure prime this is ^ (p’^ — 1). 
If p is a mixed prime, = a + fSi, the number is [or + /3' — 1). 
Example .—Modulus 5 + 2h 10 is a primitive root with exponent 
(h (5 + 2i) = (53 + 2= - 1) = 28. 
The residues of its powers are given in the table. 
N umber 
10 
13 
14 
24 
8 
22 
17 
2.5 
18 
6 
2 
20 
26 
28 
Index 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
Number 
19 
16 
15 
5 
21 
12 
4 
11 
23 
27 9 
3 
1 
Index 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 or 0 
The 6 (28) 
12 primitive roots 
are 
10 
14 8 
18 
•7 
26 19 15 
21 
11 27 
Exam 2 ole .—Modulus 7. Primitive roots have exponent <!> (7) = 7" — 1 = 48. 
2 + i is a primitive root, and the residues of its powers are given in the table. 
Number 
2 + f 
3 “b 4i 
2 + 4f 
oi 
4 + 6i 
2 + 2i 
2 4- Si 
5 
Index 
1 
.7 
3 
4 
5 
6 
7 
8 
Number 
3 + 
1 + (5i 
3 “b 6i 
i 
6 + 2^ 
3 + oi 
3 + 2^■ 
4 
Index 
9 
10 
11 
12 
13 
14 
15 
16 
N umber 
1 + 4i 
5 + 2/ 
1 + 2?: 
5i 
2 + Si 
1 + i 
1 + Si 
6 
Index 
17 
18 
19 
20 
21 
00 
23 
24 
Number 
5 + 
4 “b 3i 
5 “b 3^ 
Ai 
3 + i 
5 “b 5 1 
5 + i 
.7 
Index 
25 
26 
27 
28 
29 
30 
31 
32 
Number 
4 + 2?: 
6 + i 
4 “b f- 
6i 
1 + 5i 
4 "b Ai 
4 “b 
3 
Index 
33 
34 
35 
36 
37 
38 
39 
40 
N umber 
6 “b 3i 
2 + 5i 
6 + 5i 
2i 
5 + Ai 
6 “b 6 1 
6 “b 4?- 
1 
Index 
41 
42 
43 
44 
45 
46 
47 
0 
