FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
241 
and 
382 and 826 with exp 3®. 
Hence we may form, say, 
998. 826 = 173 (mod 999) with exp 2. 3^. 
and 
Thus 
487. 382 = 220 (mod 999) with exp 2h 3^, 
173 with exp 18 1 
220 with exp 36 J" 
generate completely the 2^, 3“^ numbers prime to 999. 
Indices and Tables of Indices. 
(32.) Let Gg . . . be a complete set of independent generators for modulus m. 
Let their exponents be 1 2 , .. . 
Then any number a prime to m is expressible in the form 
a = Gf^ Gg'- . . . GJ'^ (mod ni), 
where 
ii H ^23 
We may thus make a table giving the set of indices 4, ... 4 which correspond 
to any number a, and conversely the number a which corresponds to any set of 
indices. 
We may conveniently write 
a = (4, 4 , . . . 4), 
and then if 
Cl — {i i < 2 ,, • - • ^ k) 
we shall have 
aa' = (4 + i\, 4 + ^'g, . . . 4 + *4) 
and 
cP = ( 45 , 45 , . . . is). 
With tables of this kind for every modulus we can at once solve any congruence of 
the form 
ax’‘ = h (mod m) 
whatever m may be, if a and h are both prime to m. For suppose that 
0^ (^1 ^2 * * * ) 
h = {i\ fg . . . ) 
X — (Ij, Ig, . . . ) 
2 I 
MDCCCXCIII.—A. 
