FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
231 
Suppose that U^. To, . . . U^, are the most general set of p-power-exponent generators. 
We know already (Proposition 27) that their exponents are • • • and we may 
suppose them to be so in this order. 
Since each number U is itself a jj-power-exponent number, each is expressible as a 
product of powers of the gs. 
Let 
= (mod m) 
ro = g{^^g>g^^^ . . . 9,y- (mod m) 
= p'fV (mod m) j 
{Note. — ir, s, is the index of g,- in the value of F^.) 
Since the exponent of F^ is i9\ it follows that is the least multiplier that makes 
= 0 (mod 2^^"), 
= 0 (mod i9'^), 
&c.. 
(Proposition 8, Corollary), and similarly forp^% &c. 
We may insert here a lemma, which will be useful presently. 
Lemma.—If I,. > I,, then i^s is divisible by p. 
i,.g is the index of g,. in the product F„ 
Fj =. . . gj’’^ . . . (mod m). 
Hence (Proposition 8, Corollary) the exponent of F^ is exponent of gl'^. 
Suppose, if possible, that Is can be prime to p. 
Then the exponent of gf is p^'' (Proposition 4). 
Now the exponent of F^ isp^*, 
therefore 
therefore 
<j: p-', 
Is ^r, 
which is contrary to the supposition I < Ir, and therefore Uj cannot be prime to p. 
Hence, if I > I, then Is is divisible l^y p. 
Take any one whatever of the j9-power-exponent numbers, (modulus m). 
Then, since the numbers F are also generators of these numbers (the p-power- 
exponent numbers), the numbers gfgf . . . gl>^ (modulus m) must be expressible in 
the form FpTg"^^ . . . F^^'* (modulus m) in one way, and one way only. 
Therefore 
9\"9z" • • • = ■ . . F/M(mod m) 
