FOR COMPOSITE MODULUS, REAL OR COAIPLEX. 
199 
where 
43 has exp 2"! 
177 „ „ 3 > 
141 
(11.) The number of numbers that belong to any exponent 
a prime p is (f) (t). 
If there are any numbers w^hich have exponent t, mod let 
Then because 
a^= 1 (mod j)), 
therefore each of the t in congruent numbers 
2 8 f 
a, a , . . , 
when raised the power is = 1. 
Hence each is a root of a:'' = 1 (mod p) wdiich has only t incongruent roots. 
Therefore every number jB which has exponent t, and which is therefore such that 
= 1 is included in the above set. Hence every number with exponent t is to be 
found in the above set. Now, of the powers of a there are [t) which have their 
index prime to t, and which, therefore, have exponent t (Prop. 4). 
Therefore if there is one number with exponent t there are </* {t), and no more. 
Now if ti, . .., are all the divisors ofp — 1, and, therefore, all possible exponents 
(Prop. 3, corollary), 
^ ih) + ^ ih) + •••=</> i'P)- 
Corresponding to each value t there are either ^ (t) numbers or none with t as 
exponent. The number of numbers altogether is {p). Hence in no case can there 
fail to be ^ (t) numbers with exponent t. 
Corollary .—In particular there are (j) (P ~ 1) tiumbers with exponent p — ], 
modulus _p, i.e., any odd prime y) has (f){p — 1) primitive roots. 
(12.) The exponents to which a number a belongs for successive powers of a prime 
p as moduli. 
We suppose that a is prime to p and that p) ^ 2. 
Let the exponents to which a belongs for the moduli p, . . . be 
h’ ' ■ • respectively. 
Then because 
f/K+\= 1 (modp'^'^’), 
therefore 
oA+i = 1 (mod j/), 
t when the modulus is 
a be one. 
therefore 
divides (Prop. 2.) 
