FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
195 
Then, because 
r(fa'^'a"^" . . . = 1 (mod m), 
therefore (multiplying by cd"'*), 
akt^'a'^" . . . = (mod ?n), 
or 
a^ki'^'a'^" . . (mod m), 
or 
a^a^'a'^" . . . = a^~^ a^a'^ . . . (mod m), 
which is contrary to the supposition that no two numbers of the form cCa'"' . . . 
(mod ?n) are congruent; for the last congruence obtained shows that if we make 
0/ / // // 
r — t — s r' = 0 7'" = 0 . . • J 
where 
t — s ^ 0 (mod t), 
the two numbers are congruent. 
Hence, we must have 
5 = 0 (mod t), 
and, therefore, 
a* = 1 (mod t). 
Similarly, 
= 1 (mod t'), &c. 
Definition. —If a, a, a" . . . have exponents t, t\ t", . . . and if no two of the 
tt't" . . . numbers that can be formed by products of their powers are congruent 
modulus ni (as in the last proposition), then these numbers, a, a', a", . . . which 
generate the tt't" . . . incongruent numbers, wdll be called independent generators. 
The last proposition may then be stated thus :—If a product of powers of a set 
of independent generators be congruent to unity, then each of those powers is itself 
congruent to unity. 
(8.) If a, a, a", . . . independent generators, have exponents t, t', t", . . . then the 
exponent of aaa" ... is the L.C.M. of t, t', t" . . . 
Let T be the L.C.M. of t, t', t", . . . and r the required exponent of aaa" . . . 
therefore, 
a*' = 1 (mod m), 
= \ (mod m). 
Similarly, 
a!'^ 1 (mod m), &c., 
