280 MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
then 
ct zzi 5 -|- (8 bi) 4 -f- (8 -j- 4?-). 3 . (mod. 10), 
= 5 + 2 4- 4i H- 4+ 2^■ (mod 10), 
= \ -\- Qi (mod 10). 
(xxi.) I’he number of numbers which belong to a given exponent when the 
modulus is a power of a prime. 
We have three cases to consider 
(1) When the modulus is a power of a pure prime. 
(2) When the modulus is a power of a mixed prime. 
(3) When the modulus is a power of 1 + i. 
(1.) We saw in Proposition xiia, that for the modulus we can generate the 
<I> (j/) = residues prime to the modulus by three generators 
and 
f and f each with exp ^ 
h with exp — 1. 
How many of the residues have exponent pH, where 5 4 ^ ~ 1 and t divides 
p"- - 1 ? 
Of thep^ — 1 powers of h, (f) (t) have exponent f. 
Of the p)^ ~ ^ powers ofyi p>^ have exponent a power of p 
Of the p^~^ powers of f, p^ have exponent a power of p 4 P'’- 
Therefore f and f' generate p^^ numbers with exponent '\> and similarly 
numbers with exponent 4 P*"" 
Therefore f and f' generate p~^ — p^^ ~ “ numbers with exponent pt 
Therefore y] f, and g generate (p^" — p^^“") 9 [t) numbers with exponent ift, i.e., the 
number of numbers with exponent jjH mod p^ is (p““’ — ~ “) ^ (0- 
In particular the number of numbers with exponent p® is p-^ — p“® “and the 
number of numbers with exponent t is ff) (t) if t is prime to p. 
( 2 .) Any power of a mixed prime, (a + (Hy, has primitive roots, and hence, as in 
( 21 ), the number of numbers with exponent t [any divisor of (4 + ~ (“~ "b 
is </) {t). 
(3.) The number of numbers with exponent a given power of 2 for modulus (1 + iY 
was found completely in Propositions xiv. and xv. 
(xxia.) It will be convenient for the succeeding propositions to express the number 
of numbers which have a given exponent for modulus (1 + ?)h in terms of the 
exponents of the generators. By so doing we shall avoid the detailed discussion of 
the cases arising fi-om ditferent values of X, which was necessary in Part I. 
Suppose 2 '^, 2 "', 2 "" are the exponents of the generators. 
The exponent of the product of any powers of the generators is equal to the 
highest exponent of the tliree (Proposition viii.). 
