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

Wayne L. McDaniel

International Journal of Mathematics and Mathematical Sciences, 1990, vol. 13, 1-12

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 ) .

Date: 1990
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:894903

DOI: 10.1155/S0161171290000023

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