290 
MR.. G. T. BENNETT ON THE RESTBUEE OF PO\yERS OF NUMBERS 
and so 
« = [{y, - 1) + I.}'- [(/3, - 1) + l]-^- [{y, 
[(r I — I) + 17‘ [(y'o — 1) I'o + Ij'^ . . . (mod m). 
The quantities in the Ijrackets we shall denote by . . . g^. Each is congruent 
to unity for all but one of the principal factors of 7n as moduli. Their exponents are 
the exponents of any set of ^^power-exponent generators (Proposition xxvi.), viz., 
■p\ ’P\ • • . 
Next let 2^ = 2. 
Then take (f), (f)', generators for mod (1 + i)" 
yy a number with exp 2'^' (mod 
&c. &c. 
Then 
and, therefore, 
y\ a number with exp 2'^'' (mod Qj'"'), 
&:c. &c. 
a,) = [I’ood (1 -j- v')''] 
= yp (mod P7‘), 
&c., 
= rV'* 
&c. 
a = - 1) ^0 + 17 [(f - 1) ^0 + IF [(f' - 1) ^0 + IF' [(ri - 1) + !]'■ . . . 
[(r'l - 1) + I]*'' • . • (mod ?JI), 
and the numbers in brackets are unitary generators of the numbers, modulus m, with 
exponent powers of 2. 
Each is congruent to unity for all but one of the principal factors of wi, and their 
exponents are 2"“ 2“'° 2""“ 2"^ 2"^ .. . 2'"'^ 2''-' . . . 
Thus, in either case, whether is an odd prime or equal to 2, we can form a set of 
^i-power-exponent generators (having the exponents found to be necessary in 
Proposition xxvi.), such that each is congruent to unity for all but one of the principal 
factors of in [(1 -f i), P^S . . . QE‘ . . .] as moduli. 
(xxviii-xxxi.) Propositions 28-30 are concerned with a discussion of the most 
general mode of formation of a set of p-power-exponent generators. They are 
throughout completely applicable to the case of complex numbers and moduli. The 
result we may state as follows. Let g^, g^, . . . g^ be a set of unitary p)-power-exponent 
generators. 
Then \\ To . . . (with the same set of powers of^; as exponents) as given by 
