Generalizations of Lehmer's equations, greatest common divisor matrices, and Euler's totient for a UFD

Abstract

One of the most researched functions of number theory is the Euler -function, or totient function. The number (n) yields the cardinality of the group of unit of the ring module n, and depends on the canonical decomposition of a natural number. In Chapter II of this dissertation, we study the -function which depends on noncanonical decompositions of a natural number into a product of some of its divisors. Using the -function, we will define some equations which generalize certain well known number-theoretic equations involving the -function. These equations will be studied, and complete solutions for certain classes will be given. In Chapter III, the results obtained in Chapter II will be applied to Lehmer''s equations, and to finite rings with identity. In 1932 D. H. Lehmer considered the equations 2(n) = n 1 and 2(n) = n + 1. Lehmer found all solutions for the second equation, with n being divisible only by six or less distinct primes, and no solutions to the first equation. Since then, no solutions have been found. C. Poncrance gave an upper bound for such a solution. Results obtained in Chapter II will be applied to the second equation to give a much improved bound. As for the second equation, we will use an equation involving the -function to show that no solution divisible by exactly seven distinct primes exists. Beslin and Ligh initiated the study of the generalization of Lehmer''s equation to finite rings with identity. It will be shown that the equations involving the -function generalize the ring-theoretic equations. Beslin and Ligh noted that Boolean rings are automatically solutions to the ring-theoretic equations. The equations involving the -functions will be used to give two infinite classes of solutions, other than the Boolean rings, to two of the ring-theoretic equations. There are many generalizations of the -function. Most are number-theoretic generalizations. In Chapters III and IV we give two ring-theoretic generalizations of the -function. One is for the purpose of evaluating determinants of greatest common divisors matrices defined on a U.F.D. (unique factorization domain). The other involves the cardinality of certain groups of units.