218 
MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
First supjDOse that /c > 3, and so k > 1. 
Then there are, mod 2“' 
2'^' numbers with exponent 2"' 
C ) k '-1 
" 55 55 55 
93 
" 55 55 55 
Q 
^ 55 55 55 
1 number ,, ,, 
Hence there are 
2"' ^ numbers with exponent a power of 2 2'^' 
2^' „ „ „ „ >2*'-^ 
2 ^ „ » „ „ >2 
1 number ,, „ „ >2® 
If 
cr > K then there are 2*''^^ numbers with exponent a power of 2 2“^, mod 2''''^', 
and if 
(T K then ,, ,,2 ,, ,, ,, ,, ,, ,, 
(this holds unless cr = 0, and then there is one number (unity) with exponent l). 
In either case, then, there are ^ numbers with powers of 2 :!> 2“^ as exponents, 
where {k')^ is to be replaced by 
cr if k' > cr 
K „ K ^(T 
Next for mod P/' there are 
2'^! -1 numbers with exponent a power of 2 2*' “ ^ 
2 ^ 1-2 2*1-2 
5 5 5 5 5 5 5 5 ^1 ^ 
&c. 
Hence (using the same notation) there are numbers (mod P^^') having exponent 
a power of 2 2''. 
Thus ocq may have each of 2'^<^ values, each of 2^*'^'^ values, &c., and the corre¬ 
sponding value of a has exponent a power of 2 (mod m), 2^ 
Hence the number of these is where in %k = k -{■ -{• k. 2 • each 
number k is to be replaced by cr if it exceeds cr. 
2*’-i 
22 
2 
1 
