274 MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
The Lemmas (iii.) and (iv.) show that the group to which P' 3 ^ 3 belongs is 
one of 
t.e., 
a, / 8 , 8 , a/ 8 , a 8 , ySS, or a/BB, 
a, /3, s, y, e, r], or 9, 
and in no case is the product congruent to unity, save when each factor is so 
separately. 
The result, therefore, is that for modulus (1 + iY we have to take three generators 
with the exponents determined in the last proposition, and such that the product of 
the three numbers with exponent 2 that they separately produce shall not be 
congruent to unity. 
Examiole .—Mod (1 + i)^. $ (1 + = 2 h 
The exponents of the generators are 
23, 2^, 2l 
The 2 ® numbers with exp 2 ^ are 
1 + 2b 
2 + i, 
3 + 2^■ 1 
2 + 3?: I 
j 
(mod 4). 
The 2 ® + 2 ^ + 2 ^ numbers with exp 2 ~ are 
. > (mod 4) and 3, 5, 3 + 4b 5 + 4b 1 + 4b 7 + 4i (mod 8 ). 
3^ J 
The 23 + 2+1 numbers with exp 2 are 
15, 9, 7, 1 + 8b 15 + 8b 9 + 8b 7 + 8h 
If of these we take 
and 
we have 
and 
and 
2 + ^ with exp 8, 
3 with exp 4, 
i with exp 4, 
(2 + ?:)^ = 9 + 8f [mod (1 + i)®], 
33 = 9 [mod (1 + iYY 
=15 [mod (1 + 
9 + 8^. 9 . 15 = 15 + 8^ ^ 1 [mod (1 + i)®]. 
therefore, 2 + i, 3, and i generate the 2^ numbers. 
