262 
MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
(xiii.-xvi.) Propositions (13) to (16) are concerned with the moduli 2^. In Proposi¬ 
tions xiii.-xvi. we shall investigate the moduli analogous to these in the case of 
complex numbers, viz. (1 + i)h [Propositions xiii.-xvi. will not be made to correspond 
individually to Propositions (13)-(16) ; they are intended to cover the same ground.] 
(xiii.) In the case of the smaller values of X, from 1 to 7, we shall find results, some 
of which, though they are really particular cases of the general results for any value 
of X, are not conveniently included under them. 
We shall start by a separate treatment of each of the first seven moduli, finding— 
(i.) the numbers which belong to each exponent; 
(ii.) what numbers must be taken in order to generate the complete set of residues. 
Mod 1 + i. There is only one residue, viz., 1, whose exponent is 1. 
Mod (1 -p f)~. cf> (1 -|- f)^ = 2. There are two numbers, viz.. 
and 
i with exp 2 
1 with exp 1 
Thus, for this modulus, ^ is a primitive root. 
Mod (I -j- f)®. cp ( 1 -p i)3 4. The four residues are— 
i, 2 + ^ with exp 4, 
3 with exp 2, 
1 with exp 1. 
i and 2 i are both primitive roots. 
Mod (1 + ^)b (1 iy = 8. The 8 numbers with their exponents are 
Exp 4. 3f, 2 + b- 2 + 3b 
Exp 2. 3, 1 -p 3 -p 2i, 
Exp 1. ]. 
The square of each of the four numbers with exponent 4 is congruent to the same 
number with exponent 2, viz. 3. 
Hence, to generate the 8 numbers, we must take any one of the four numbers with 
exponent 4, and either of the two numbers with exponent 2. viz. 1 + 2q 3 + 2b 
Mod (1 + 0^- ‘h (1 + f)^ = 16. The 16 numbers, with their exponents are 
Exp 4. i 3f 2 -p ^ 2 + 3f 4 + f 6 + « 4 -f 3f 6 + 3b 
Exp 2. 3 5 7 1 -p 21 3 -p 2i 5 -p 2i / -p 2i 
Exp 1. 1. 
