266 MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
Numbers ivith exponent 2 mod (1 + iY. (X> 4.) 
All these numbers, together with 1, which has exponent 1, satisfy the congruence 
= 1 [mod (1 + ; 
therefore 
a = i 1 [mod (1 -r 0^ ~ T 
gives (together with unity) the numbers which have exponent 2, mod (1 + 
We get thus 7 numbers with exponent 2, mod (1 + iY. They are congruent to 
— 1 
±l + (l + ty-2 
± 1 + (1 + 
± 1 + (1 + ,)^-2 + (i 
l' [mod (1 + iy]. 
J 
The product of any two of these is congruent to a third. If we name the 
7 numbers thus— 
a = — 1, 
= + 1 + (1 + 
r = - I + {i+iY~\ 
S = + + 
e=-l+(l+^y“\ 
y)=+i + (i + + (1 + ^y-^ 
^ =-! + (!+7)^-2+(1 + ^y-b 
then these relations may be written thus— 
“/3r = 1, 
«Se = 1, 
0.7)6 = 1, 
^St] = 1 
y8e^ = 1, 
ySO = 1, 
y^T] = 1, 
[which includes aj8 = y, ay = /3, /3y = a, because = 1, y8~ = l, y' = l]. 
Numbers ivith exponent 4, mod (1 + 7)\ {X> 6.) 
The numbers which satisfy 
cd z= I [mod (1 + ^ )^], 
