216 MR. G. T. BERNETT ON THE RESIDUES OF POWERS OF NUMBERS 
for mod 2", all 2'‘“^ numbers nave powers of 2 as exponents, 
,, (ji (2''') + (f) (2''>“‘) +... + 1 = 2*' have powers of 2 as exponents, 
„ P/% (2-) + (2«--‘) + . . , + 1 = 2- 
&C. &C. &C. 
Hence, in all, there are numbers, which have unity or a power of 2 as 
exponent. 
Note. — Any number a, mod m, with any exponent, must be congruent to a product 
of one number from each set of •• • numbers, with powers of 2 as exponent, p~^ 
with powers of u as exponent, . . . &c. (Proposition 10.) 
Hence in all we get from these 
^ _ numbers, 
= <^ (2-) ^ (P,0 . . ■ 
= (f) [m) numbers, i.e., the complete set of numbers prime to m. 
(23.) The number of numbers having exponent p®, mod m, if being a divisor of the 
greatest exponent. 
Let a be such a number and let 
« = ao^o + '^1^1 + + • • • 
then the exponent of 
mod 2* 
a, mod Pd' I 
^3 mod Po^- I 
&c. J 
must each be unity or some power of p, and the greatest power of p must be p 
(Proposition (9), Corollary); p being an odd prime olq is necessarily unity. 
There are 
(f) (y)'‘) numbers, mod P,^^* with exp y)'', 
U 3 J 
P 
h-\ 
kc 
&c,. 
and therefore there are 
jD*' numbers with exp a power of p < p'', 
P' 
<P’' 
l'cC. . 
&c. 
5) 
3) 
