International Journal of Mathematics and Mathematical Sciences
Volume 13 (1990), Issue 1, Pages 13-24
doi:10.1155/S0161171290000023
An analogue in certain unique factorization domains of the Euclid-Euler theorem on perfect numbers
Department of Mathematics and Computer Science, University of Missouri—St. Louis, St. Louis 63121, MO, USA
Received 14 July 1988
Copyright © 1990 Wayne L. McDaniel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We show that there exists a natural extention of the sum of divisors function to all unique factorization domains F having a finite number of units such that if a perfect number in F is defined to be an integer η whose proper divisors sum to η, then the analogue of Euclid's theorem giving the sufficient condition that an integer be an even perfect number holds in F, and an analogue of the Euclid-Euler theorem giving the necessary and sufficient condition that an even integer be perfect holds in those domains having more than two units, i. e., in Q(−1) and Q(−3).