FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
223 
(25.) Suppose G], Gg, G 3 , . . . with exponents ti, . are capable of acting as 
complete generators of the ^ (w) numbers. 
Then 
(i.) . . . = (f) (m) 
and 
(ii.) the generators must be independent. 
Since the numbers G generate the complete set of <f){'}n) numbers they can 
generate in particular the numbers which have a power of a prime p (which is a 
factor of (f) (m)) as exponent. 
Suppose the highest powers of p which occur In . . . are p'\ . . . respec¬ 
tively. 
Say 
h = n" 
&c. 
Then we can express the numbers G thus (Proposition 10) :— 
G^ = (mod m), 
G 3 = (mod m), 
&c., 
when g^ has exponent and has exponent i(\, &c. 
Suppose now that 
Gf^Gg’^Gs’^ ... is congruent to a number with a power of p as exponent, 
and so 
• • •) • • •) 
gi, and therefore gp, has a power of p as exponent (Proposition 4), and so for gp, &c. 
Therefore 
a power of p as exponent. 
The exponent of h^, and therefore of hp, is prime to p, and so for hp, . . . &c. 
Therefore the exponent of hphp . . . which divides the L.C.M. of the exponents 
of hp, hp, ... is prime to p>- 
Therefore the exponent of {g{pp . . .) {Jiphp . . .) is the L.C.M. of a power of p 
and of a number (the exponent of hphp . . .) prime to p. Now the exponent is to be 
a power otp. Hence the exponent of hphp . . . must be unity. 
Therefore 
hphp . . . = 1 (mod m). 
