2G8 MR. G. T. BERNETT ON THE RESIDUES OF POWERS OF NUMBERS 
Hence the numbers with exponent 8 are 
± 1, ±i,+(n-in‘ 
\\-5 
± 1, ± i, + (1 + *■)' 
±1. ±i +(1+ *T-* + (i+»■U^ 
> mod (1 + i) 
A.-4 
In nil mil er they are 
4 X 3 X N (1 + ^■)^ 
= 2h 3.2^ 
= 2® + 2h 
We have not specially examined the modulus (1 + ij'. The general results 
already obtained give us the numbers with exjionents 2 and 4. We will find the 
numbers with exponent 8. 
Of the solutions of 
[mod (1 + 
we must exclude those of 
a^= 1 [mod (1 + 'i)®]. 
a® = 1 [mod (1 + •2^)®] 
gives 
a'' = 1 [mod (1 -f- ^)®] 
and 
= d: 1 [mod (1 -T 
and therefore 
3i, 2 d“ ^3 2 d” 3'?', 3, 1 -f- 2i, 3 d~ 2?-, L [mod (l d~ [See mod (1 -fi ^)^*J 
The numbers to be excluded are 
i.e., 
= 1 [mod (1 d- '2')®] 
= dr 1 [mod (1 -f 
« = dz I3 ± 'i [mod (1 d- 
a = 1, 3, b 3i [mod (1 1 )^]. 
Hence the numbers = 2 -f- b 1 + 2/, 2 d- St, 3 -f 2t [mod (1 -fi Imve exponent 8. 
The number of them is 4 X 2“^ = 2®. 
