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
We show that there exists a natural extention of the sum of divisors function to all unique factorization domains having a finite number of units such that if a perfect number in is defined to be an integer whose proper divisors sum to , then the analogue of Euclid's theorem giving the sufficient condition that an integer be an even perfect number holds in , 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 and .
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