FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
221 
g-^ be a primitive root of 
ft ^ ^2 
&c. &c., 
and let 
P '0 have exp 2'^““ mod 2'^ 
y be = 2'^ — 1 or 2''"^ — 1 mod 2'‘ 
(In the case when k = 2,/does not occur, when k = 1 or 0, neitherynor gQ.) 
Then 
and 
Therefore 
= g^^ (mod 
= gJ^ (mod Pg^-) 
&c., 
> sav. 
*/ 
a = g^'>fJ^Q + + . . . (mod m) 
= (/^o + + • • -V (i/o^o + + • • •)'“ (4 + 9i^i + • • •)'' • • • 
= [(/- 1) ^0 + 1? \.{9o - 1) ^0 + IP \i9i - 1) + If ■ ■ . (mod m), 
where 
j is referred to mod 2, 
h „ „ mod 
ii „ „ mod fji (Py), 
&c., 
and so the set of indices j, . . . correspond uniquely to the number a: and the 
numbers 
(/~l-)^o+P (S'o ~ 1) ^0 + I; (i/i ” 1) + P • • • 
are a set of independent generators, with exponents 
2, 2-“, (Pi*'). • • ■ 
which generate completely the (f) [rn) numbers, mod m, which are prime to m. 
Example .—Take the modulus 
m = 112 = 2b 7. (j) (w) = (23) (2. 3) = 2b 3 = 48. 
If 
a = olq (mod 2'^), 
