FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
281 
The generator with exponent 2'' generates 
2® numbers with exp :|> 2* if /c > 5, 
2'‘ numbers with exp 2^ if k^s. 
Suppose by {k)^ we denote that k is to be rej^laced by s if k exceeds s. Then in 
either case the generator with exponent 2" generates 
2^'"^® numbers with exp ^ 2\ 
Similarly the two other generators generate respectively 
2('‘'t numbers with exp 2\ 
and 
2 (*"^3 numbers with exp 2\ 
Therefore the three generators generate 
2 fK + k' + -c'')s with exp ^ 2^, 
i.e., thei’e are, for modulus (1 + iy, 
2 (k + k' + /t")s numbers with exp ^ 2t 
Similarly there are 
2 (>t + «' + *")s_i numbers with exp 2®“h 
Hence there are 
(k + k' + C} (k + k' + K '}g _ 1 
numbers with exponent 2^ for modulus (1 4- iy. 
This result clearly holds good when one or two of the k’s is absent, as is the case 
when X is less than 5. 
(xxii.) The number of numbers, for modulus m, each of which has as exponent some 
power of a prime 2^ being a divisor of {in). 
Let 
m = (I +^■)'‘P/‘P2^ . . . . . . 
where P^, Pg, . . . are pure primes and Q^, Qg, . . . are mixed primes. 
^ (m) = tP (I + $ (P/i). <E) (Pg^^) . . . (P (Qg^^) . . . 
(J) (1 + z= 2''“2''»'2'''’" 
where 2''“, 2"“', 2“^", are the exponents of the generators for mod (1 and suppose 
that 
cp (P/i) = {'Py — 1) = . . . 
Cp (Pg^^) = P22(A,-I) _ 1) — _ 
&c. &c. 
2 o 
MDCCCXCIII.—A. 
