FOR ART COMPOSITE MODULUS, REAL OR COMPLEX. 
217 
Hence-if 
< s there are numbers, mod with exp a power of :j> 
and if 
li ^ s there are p' numbers, mod with exp a power of pt p)' 
Suppose that (?i)s stands for s if > s and for if l-^ < s, then in either case 
there are numbers, mod P^^^, whose exponents are powers of p) pf. 
Similarly there are numbers mod P 3 ^S whose exponents are powers of p ;J> p^ 
and so for P^^^ &c. 
Giving any one of these values to a^, values to &c., we obtain 
^Ci). + (y. +.. ■ — pirn numbers, mod rn, 
whose exponents are powers of p j> p>^; where in p^^'‘^’ each number Z in + Zg + • • . 
is to be replaced by s if it exceeds s. 
Similarly, there are numbers, mod m, whose exponents are powers of p) :j> p'~^ 
&c,, &c., numbers with exponents jo or unity, and 1 number with exponent 1. 
Hence the number of numbers whose exponent is p% mod m, is 
p(2»> _ which when s = 1 is p)^-’^^ — 1. 
Corollary .—If p^ be the higlrest power of p which can be an exponent for mod m, 
i.e., if jT is the highest power of p that divides the greatest exponent, i.e., if s is the 
greatest of the numbers Z^, Zg, . . . then 
1 + - 1 ) + + . . . + 
is the number of numbers having as exponent (mod m) a power of p (or unity) as 
exponent. 
Now since s is equal to the greatest of the quantities Z^, Zg, . . . (or rather is not 
less than any) [Xl)^ = XI and therefore p^'^’’^’ = p)^^: the result which was obtained in 
the last proposition. 
(23a.) To find the number of numbers having exponent 2°’, mod m. 
(It will be convenient to write now k = /c' + 2, so that ra = 2'^'+ ^P/'Pg^^ . . .). 
If a be any such number, and 
the exponents of 
a = + . . . (mod m), 
ccq mod 2* 
mod P^^i 
ag mod Pg^” 
1 
I 
must each be unity or a power of 2, and the greatest power of 2 must be 2X 
MLCCCXCIIl.— A. 2 F 
