Abstract
The present note aims to exhibit some elementary relations between wellknown methods of finding the primitive roots of a number and the properties of the cyclic group. Incidentally we arrive at a fundamental theorem relating to the primitive roots of a special class of numbers. A corollary of this theorem gives the primitive roots of all the prime numbers of the form 2p + 1, p being a prime, while it has been customary in the works on the theory of numbers to devote two theorems to the primitive roots of such prime numbers.* The note has close contact with the paper published in this JOURNAL under the title Some Relations between Number and Group Theory and may be regarded as a continuation of this article.t It is known that the necessary and sufficient condition that a number g has primitive roots is that the cyclic group G of order g has a cyclic group of isomorphisms I. The numbers which are less than g and prime to it may be made to correspond to the operators of I, unity corresponding to the identity, in such a way that I and the group formed by these numbers, when they are combined by multiplication and the products reduced with respect to modulus g, are simply isomorphic. The orders of the operators of I are the indices of the exponents to which the corresponding. numbers belong. In particular, g -1 corresponds to the operator of order 2 and the primitive roots of g correspond to the operators of highest order in L Hence the method of finding the primitive roots of a number is equivalent to that of finding the operators of highest order in a cyclic group. One of the most instructive methods for finding all the primitive roots of g is analogous to the method known as the Sieve of Eratosthenes for finding
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