284 
MR. G. T, BENNETT ON THE RESIDUE.S OF POWERS OF NUMBERS 
Tlie mimber of numbers with exponent 2 
Tims there are 
_ 2(2 +1 +1 + ^). _ [ 
— .>!• 
1 . 
2 ^ — 2^' numbers with exp 4. ' 
2^-1 „ „ 2 . > 
1 
55 
55 
1 . 
Corollary to xxiii. and xxiha.—The number of numbers that belong to any 
exponent is simply the product of the number of numbers that belong to each of its 
principal factors as exponents. 
Thus, for mod m, tlie number of numbers with exp t = p^<f ... is 
Example. —Mod (1 + . 3“ (3 + 2i) (4 + i) = -- i. 72 (10 + IH). 
cp (1 + = 2. 2b 2b 
cp ( 32 ) = 2b 3. 3. 
(h (3 + 27) = 2b 3. 
(4 + 7) = 2b 
The highest exponent is 2b 3^. 
cp (?)i) = 2ib 3b 
The numbers k are 1. 2. 
Therefore 
(20i= 14, 
(203= 13, 
( 203 = 11 , 
(2k)i = 6, 
(% k )„ = 0. 
.>ir 
013 
numbers have 
exp 
or 
> 
Ol3 _ 
Oil 
03 
55 
55 
5 5 
Oil_ 
06 
o3 
55 
55 
55 
^ 5 
06 _ 
1 
55 
5 5 
5 5 
0 
The numbers I for the prime 3 are 1. 1. 1. 
