286 MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
where the numbers in the square brackets are a set of generators whose exponents 
are, in order, Kq, k'q, k"q, — 1), P/'“h . • . (Qi'"') • • • whose product is equal 
to $ (m). 
Example. —Mod 10 = — (1 + (2 + i) (1 + 2 i) 
1 with exp 2 generates the 2 numbers mod (1 + i)~ 
2 ,, 4 „ 4 „ 2 + t 
2 „ 4 „ 4 „ 1 + 2h 
Hence the following will generate the <1>(10) = 32 numbers, mod 10, viz., 
where 
ili+ fa 
f, + 2f,+ ft 
^1 H ~ ^3 + 2^3 
> mod 10, 
^ 1=5 
Hence 
= 8 +6?' 1^3 = 8 + ii. (See Example, Proposition xix.) 
5/+ 8+6t + 8 + 4?;=6 + 5r 
5 -|- 2(8-l“b'i)-|- 8 -p 4? 9 -p >(mod 10). 
5 + 8 + 6? + 2 (8 + li) = 9 + 4i^ 
9 -p 4^ with exp 4 
9 + 6i „ 4 
>generate the 32 residues. 
6 + bi 
The indices corresponding to each of the 32 numbers are given in the following 
table. 
Numbers. 
Indices of 
Numbers. 
Indices of 
6 + 54. 
9 + 64 . 
9 + 44. 
6 + hi. 
9 + 64 . 
9 + 41. 
1 
0 
0 
0 
G + 54 
1 
0 
0 
9 ^ a 
0 
0 
1 
4 + 9i 
1 
0 
1 
5 + 2i 
0 
0 
2 
7i 
1 
0 
2 
7 + 8i 
0 
0 
0 
0 
2 + 34 
] 
0 
3 
9 + 6i 
0 
1 
0 
4 + 4 
1 
1 
0 
7 
0 
1 
1 
2 + 54 
1 
1 
1 
3 F 8 'i 
0 
1 
2 
8 + 34 
1 
1 
2 
5 + 44 
0 
1 
3 
94 
1 
1 
3 
5+84 
0 
2 
0 
34 
1 
0 
u 
0 
3 + 24 
0 
2 
1 
8 + 74 
1 
2 
1 
9 
0 
2 
2 
4 + 54 
] 
2 
2 
1 + 64 
0 
2 
3 
6+4 
1 
2 
0 
0 
7 + 24 
0 
3 
0 
2 + 71 
I 
»'> 
0 
0 
5 + 64 
0 
3 
1 
i 
1 
3 
1 
1 + 44 
0 
3 
2 
6 + 94 ’ 
1 
3 
2 
3 
0 
3 
3 
8 + hi 
1 
3 
3 
E.q., 6 + / = (6 + 5/) (9 + (\i)~ (9 d- 4^)^ (mod 10). 
