FOR ANT COMPOSITE MODULUS, REAL OR COMPLEX. 
219 
Similarly the number of numbers whose exponent, mod m, is a power of 2 2'" * is 
2(2-=)o. _ 1 + 
Hence the number of numbers with exponent 2“" (mod m) is 
2(2k)o. +1 -1 
This holds for cr > 2. When cr = 1 the number of numbers with exponent 2 is 
2(2«)i+1 _ 
Secondly suppose that k = 2, and so k = 0. 
Then, if cr is > 1, may be either 1 or 3, two values. 
Hence the number of numbers with exponent 2'^ is 
2(2'£)o. +1_ “*'*'*> 
which agrees with the above when k is omitted. 
If cr = 1, the number of numbers with exponent 1 or 2, mod 2^, is 2. 
Therefore the number of numbers with exponent 2, mod m, is 
2(2k)i + 1 _ 2 
which agrees with the above when k is put = 0. 
Thirdly, let k = 1 . Then must be unity and the number of numbers with 
exponent 2"^ is where ^/< = Kj + /Cg + . . . 
Fourthly, when (/c = 0) m is odd, then again the number required is 
To collect the results ;— 
When 
or 
And when 
22p^^ip^A, _ _ _ 
or 
m = 2P;^'^iP.A^ . . . 
Pi^^Pg^^ . . . 
there are — 1 numbers with exp 2, 
2 (^9 (t+i — 2 ( 2 '')o— 1 +^ numbers with exp 2L 
there are 2'^^''^' — 1 numbers with exp 2, 
2 ( 2 K)<r _ 2 ( 2 '')<r-i numbers with exp 2°". 
(23b.) The last two propositions give us the number of numbers belonging to any 
exponent t = . , . 
For the complete set of these numbers is formed by taking all possible products of 
a number with exponent 2®, one with exponent p^, &c. 
Hence the number of numbers with exponent t is the product of the number 
of numbers that belong to each of its principal factors as exponent. 
2 F 2 
