FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
215 
A.ny-number has for its exponent, modulus m, the L.C.M. of its separate exponents 
for moduli 2% . . . the principal factors of (Proposition (9), Corollary.) 
Hence, when the exponent, modulus m, is a power of a prime p, the exponent for 
each of the principal factors of m as moduli, must each be either unity or some power 
ofy>. 
Conversely, if we take a set of numbers a^, a^, a^, . . . one for each of the moduli 
T, Pi^*, . . . such that the exponent of each, for its own modulus, is unity or some 
power of p, then the number a = + • . . (mod m), will have a power of p as 
exponent, modulus m. 
The numbers ^ are given by 
^0 = 1 (mod 2'^), 
= 0 ^mod 
^1 = 1 (mod P^^’), 
= 0 (mod ~}j, 
&c. 
Thus, by giving to the a’s all possible sets of values consistently with each having 
unity or a power of p as exponent, we shall obtain all the numbers (mod m) which 
have (unity or) a power of p) as exponent. 
There are 
(f) (jp) numbers .which have exp jp, mod P^^'. (Prop. 21.) 
4 > (^P-') „ „ „ 
I &c. 
Hence there are (f) {p^^) + 4> (p''^"^ \ — p)'^ numbers, mod Pj^', whicii have 
unity or a power of p as exponent. 
Similarly, there are jP" for mod Pg^', and so on. 
Hence, in 
a = + • • • (mod m), 
we can give 1 value (viz., unity) to 
any one of values to 
„ P^^ „ ao 
&c., &c. 
and then a has unity or a power of p> as exponent, mod in ; moreover, in this way, all 
such numbers are obtained. 
We thus obtain + + iiicongruent numbers, each of which has a power of 
p (or unity) for exponent. 
In the case when pj ■= 2 
Otr 
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