204 MR. G. T. BENN'ETT OX THE RESIDUES OF POWERS OF XU:\[BERS 
therefore 
therefore 
exp of 41 = 2^ ® = 2h 
(14.) The numbers which belong to exponent 2 for modulus 2 * (k > R). 
Let a have exponent 2, modulus 2". 
Then 
= 1 (mod 2'‘) 
and 
The congruence 
gives 
a ^ 1 (mod 2''). 
cr = 1 (mod 2*) 
[a — 1) (a -f- 1) = 0 (mod 2''). 
Since a is odd, if 2^ is the highest power of 2 that divides « — 1 and 5 > 1, then 
the highest power of 2 that divides a + 1 is 2, and vice versd. 
Hence either 
« + 1 = 0 (mod 2''“^), 
or 
ct — 1 = 0 (mod 2"“’). 
Therefore (excluding a — 1 (modulus 2")) we have three numbers whose exponents 
are 2 for modulus 2^ viz., 
Ok — 1 
+ L 
1 , 2 
Ok - 1 
1. (mod 2"). 
The product of each pair of these is congruent to the third. 
( 2 '‘-‘+ 1 ) ( 2 ''— 1 ) = 2 ^-‘ — 1 1 
( 2 -^ — 1) (2"-' - 1)= ~ 2-'-' + ] =2"-' + 1 (mod 2*). 
(2"-* - 1) (2''-* + 1) = 2'' - 1 J 
(15.) The numbers which have exponent 2^^ for modulus 2*. 
We have already seen that s 4^ /c — 2, and we have already treated (Prop. 14) of 
the case when 5=1. 
Let a have exponent 2^ 
The exponents to which it belongs for successive powers of 2 as moduli are given 
either by 
exp 1. 1. 1. . . 1 2. 2b ... 2'^ 
mod 2. 2b 2^. . . . 2'‘~®. h ... 2'' 
