FOR ANY COMPOSITE MODEL CIS, REAL OR COMPLEX. 
259 
Suppose 
y = 1 + (mod ’p^) a ^ 0 (mod p), 
j'=l + ol'p (mod p') a ^ 0 (mod p)). 
The p^~^ X p^~^ {p^ — 1) numbers generated by products of powers of /‘and g will 
be the p^’'^~^'‘ [p^ “1) residues prime to the modulus jp, provided that no two of them 
are congruent. We sliall now show that no two can be congruent provided that h is 
a complex and not a real number, h being determined by the congruence ha' = a 
(mod p). 
Suppose that two of the numbers generated are congruent, say 
= (mod jP), 
where 
% ^ i (mod y , p^ — i), 
j (mod p"-'). 
This is equivalent to 
9‘" =/■''" (mod p^), 
where 
i" = i — i' Q (mod p^~^ . pr — 1), 
j"=j - / ^ 0 (modp"-i). 
J 
may be divisible by a power of p. 
Suppose 
r = 
where /3 is prime to p, and s may be any one of the numbers 2, 3. . . . X. 
Then 
fj" has exp^yb (Proposition iv.), 
therefore 
therefore 
where is prime to p. 
and, therefore, since 
g"" has exp jP \ 
(Proposition iv.) Therefore 
- l)iA - s ^ - s 
= 1 (modp^). 
Ptaise both sides to the power p^ then 
(mod j/). 
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