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

• Main Author: Wayne L. McDaniel
• Format: Article
• Language: English
• Published: Hindawi Limited 1990-01-01
• Series: International Journal of Mathematics and Mathematical Sciences
• Subjects: sum of divisors; perfect numbers; unique factorization domain.
• Online Access: http://dx.doi.org/10.1155/S0161171290000023

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).