IV. On the residues of powers of numbers any composite modulus, real or complex

The present work consists of two. parts, with an appendix to the second. Part I. deals with real numbers, Part II. with complex. In the simple cases when the modulus is a real number, which is an odd prime, a power of an odd prime, or double the power of an odd prime, we know 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 be laid down a set of numbers which will here be called g, the products of powers of which give the complete set of residues prime to the modulus.