FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
227 
Lastly^ when k = 0 or 1, w^e have 
and therefore 
(S«)i = (2b)i, 
(^k)2 = (Xu)2, 
&C,, 
and so the numbers 7i are identical with the numbers k. 
Hence the exponents of the 2-power-exponent generators are 
9<i 9*, 9*3 
-J , -J 5 - ... 
We can now see the least possible number of numbers G that can generate the 
complete set of (m) numbers. 
Since each number G contains not more than one generator from each of the 
subsidiary sets of generators as a factor, it follows that there cannot be less 
generators G than the number of generators in that subsidiary set which contains 
most. 
Now, since = — 1), and P^ is odd, therefore P| — 1 is even. 
Hence = 1 at least. Therefore of the generators of the 2-power-exponent 
numbers there are at least as many as there are odd principal factors P/s Pg^"*, . . . , 
in m : and so the number of 2-power-exponent generators is never less than the 
number of generators in any other subsidiary set. 
Therefore when 
or 
m = P/‘, . . . 
2.PA . . . 
the least number of generators G is the number of primes P^, when 
the least number of generators G is the number of primes P;l 5 + 1? when 
7)1 = 2'''"^"P/‘, . . . 
the least number of generators G is the number of primes P^, + 2. 
(27.) We shall next form a set of j^^-power-exponent generators of a particular 
kind, similar to the complete generators of Proposition (24). 
Let a be any p-power-exponent number 
a — + • • • (mod m). 
2 G 2 
