[ 'Sfl 1 
lY. On the Ue^idvc^ of Poiverii of N'OAnher:^ for any Compoi^ite Moduhtft, Real or 
Complex. 
By Geoffrey T. Bennett, B.A. 
Coiiirvinicafed hy Professor Cayley, F.R.S. 
Roccircd April 8.—Read iMay 5, 1802. 
The present Avork consists of two parts, with an a})penflix to the second. Part F. 
deals with real numfjers, Part II. Avith complex. 
In the simple cases when the modidus is a real number, udiich is an odd prime, a 
power of an odd prime, or double the power of an odd prime, we l^now^ that there 
exist primitive roots of the modulus ; that is, that there are numbers whose successive 
powers have for their residues the complete set of numbers less than and prime to 
the modulus. A primitive root may be .said to generate by its successive powers 
the complete set of residues. It is also known that, in general, when the modulus 
is any composite number, though primitive roots do not exist, there may lie laid down 
a set of numbers which will here be called generators, the products of powers of which 
give the complete set of residues prime to the modulus. 
The principal object of Part I. is to investigate tlie relations wdiich must subsist 
among any such set of generators; to determine the most general form that they can 
take; to show how' to form any such set of generators, and conversely to furnish tests 
for the efficiency, as generators, of any given set of numbers. Other results which 
are obtained as instrumental in effecting these objects, such as the determination of 
the number of numbers tliat belong to any exponent, may also possess independent 
interest. 
The object of Part II. is to make, for complex numbers, an investigation which 
.shall be as nearly as possible parallel to that of Part I. for real numbers. Much of 
the work of Part I. may be applied immediately to complex numbers ; of the rest 
some will need slight modification, and some will need replacing by propositions 
leading to corresponding results. Of those cases wliicli thus call for Independent 
treatment, the most noticeable is that of the modulus (1 + if, which is the 
complex analogue of the real modulus 2\ 
The work is put in the form of a series of propositions, and is started almost from 
first principles. The early part is consequently elementary, but the advantages of 
completeness and ease of reference may be more than sufficient to compensate for this. 
A large number of illustrative examples are given. Tliese will sometimes, perhaps, 
8.3.93 
