FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
289 
The result may be stated thus :— 
If n be the number of different prime factors of m = (1 + . , , 
excluding (1 i), then the least number of generators is 
3 if /c > 4, 
n + 2 if /c = 4, 
w + 1 if K = 3, 2, 1, 
n if K = 0, 
unless one of the primes divides Pp — 1 , Pg- — 1 , . . . and (Q 3 '''), ... in 
which case it is + 1 . 
(xxvii.) The formation of a set of p-power-exponent generators for mod vi. 
Let a be any j^-powmi’-exponent number, and 
a = a,j^o + + . . . + ^ 0 + . . . (mod m). 
Then a^, . . . a.\, . . , must each have as exponent, for its own modulus, 
either unity or a power of p. 
Suppose first that p is not = 2 , but an odd (real) prime. 
Take 
with exp p^'- (mod Pi"^'), 
73 „ „ (mod Pg^^), 
&c., &c., 
if in any case p) is not a factor of <4> (P^) y is = 1 . 
If for one of these, say Pj^, p = P^^, then take also with exponent p'\ modulus 
Pj^^, as in Proposition xii., so that and y^ each with exponent — generate 
all the j 9 -power-exponent numbers, modulus P/‘. 
Take also 
y\ with exp mod Qff', 
7 2 5? 5 55 5 
&c., &c. 
Then we have 
ao = I [mod (1 + f)"], 
and can put 
0^1 =yP/3/i (mod P/^), 
^3 = ya'^ (mod Pa^^), 
&c., &c., 
S y'i''> (mod Qp'), 
a '3 = (mod Qa'^-), 
&c., 
2 P 
MDCCCXCIIT.-A 
