FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
287 
(xxv.) Proposition (25) is completely applicable to the case of complex numbers 
and moduli. The result arrived at we may re-state as follows:—The most general 
set of numbers which generate the <3? [m] residues for modulus m must always he con¬ 
structed in this manner, viz., we must form 
a set of jj-power-exponent generators 
a set of (^-power-exponent generators 
&c. &c. 
2L <1, . • . being the prime 
factors of (p (m). 
each generator must then be formed by taking numbers from these sets, not more 
than one from each, and forming their product. Each number, moreover, in these 
subsidiary sets is to appear once and once only as a factor of one of the generators 
that are thus formed. 
It remains to investigate the most general mode of formation of a set of yi-power- 
exponent generators. 
(xxvi.) The proof of Proposition (26) holds good for complex numbers and moduli. 
It shows that the exponents of any set of ^j-power-exponent generators must be the 
same set of powers of ]) as those which occur in the $’s of the principal factors of m ; 
i.e., they are what have been denoted by . . (see Proposition xxii., 
in which the convention stated must be strictly attended to). The convention of 
Proposition xxii. makes the treatment of the 2-power-exponent numbers uniform 
with that of the p-power-exponent numbers; hence the same result is true for the 
2 -power-exponent numbers, viz., that the exponents of any set of 2-power-exponent 
generators must be 2''“, 2*'“, 2'"''", 2*‘, . . . 2*'' , , , 
The least number of generators for a (given modulus m. 
As in (26) we see that the least number of generators is equal to the number of 
terras of that row which contains most among the following :— 
Kq q q (^2 ... K 2 ^ ' 
I I V r 
'1 ' 2 -- - ^1 ^2 • • • 
m.2 . . . m\ m^ • • • 
Consider first the set of numbers, k'q k"q ... k ^ . 
Since P^ is an odd prime, therefore 
is even, and therefore /Cj always occurs. 
Similarly in 
(Q-) = (a,^ + /3r - 1) 
