FOR ANT COMPOSITE MODULUS, REAL OR COMPLEX. 
225 
number jwith exponent a power of p, the product of the remaining factors being 
necessarily raised to such a power that lip = 1 (mod m). 
(It will be convenient to call the numbers which have a power of p as exponent, the 
“ p-power-exponent numbers 
Suppose then that we take the generators G, and express each as a product in the 
manner of Proposition 10^ Then take from each the factor (if any) which has a 
power of the prime p as exponent. Then, since the numbers G can generate all the 
numbers, modulus m, wliich have a power of p as exponent, and, since in the number 
G the factor g is alone effective in so doing, it follows that the set of numbers g 
generate completely the p-power-exponent numbers. 
If we take from each generator G the factor which has a power of any other prime 
g as exponent, we obtain a set of numbers g' , which generate the g-power-exponent 
numbers, and so on for each prime which divides (ru). 
We see now, that any set of complete independent generators, G, must be formed 
from these special sets of generators; the formation of each G being effected by 
taking one number (which may be unity) from each set and forming their product. 
In order to obtain the most general set of generators, G, we have now only 
to obtain the most general method of producing each of these subsidiary sets of 
generators. We may then combine them as products (one from each set) in any 
manner we please. 
(26.) Suppose that U^, Fg, . . . independent generators, generate completely the 
p-power-exponent numbers. 
The exponent of each number, F, must be some power ofp. 
Let them be p”s p"^, . . . respectively. 
One condition that the numbers, F, must satisfy is that the number of numbers 
they generate, which belong to any power of p, p* as exponent, should agree with the 
number already found. (Proposition 23.) 
Consider any number generated 
a = . . . (mod m). 
Since F^, Fg, . . . are independent generators, therefore the exponent of a is the 
greatest of the separate exponents of Fp, Fg*'^, . . . (Proposition 8. Corollary). 
Now F^ has exponent p™^, therefore of the numbers of the form Ff ‘, there are 
p”^ with exponent a power of p p^\ 
&c., 
and similarly for each of the others, Fg, Fg. . . . 
Hence the number of numbers of the form a which have, as exponent, a power of 
V ^ pi isp®”^' (using the notation of Proposition 23). 
MDCCCXCIII.—A. 2 G 
