International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 1990 / Article

Open Access

Volume 13 |Article ID 894903 | https://doi.org/10.1155/S0161171290000023

Wayne L. McDaniel, "An analogue in certain unique factorization domains of the Euclid-Euler theorem on perfect numbers", International Journal of Mathematics and Mathematical Sciences, vol. 13, Article ID 894903, 12 pages, 1990. https://doi.org/10.1155/S0161171290000023

An analogue in certain unique factorization domains of the Euclid-Euler theorem on perfect numbers

Received14 Jul 1988

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).

Copyright © 1990 Hindawi Publishing Corporation. 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.


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