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 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...
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| Format: | Article |
| Language: | English |
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Wiley
1990-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171290000023 |
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| author | Wayne L. McDaniel |
| author_facet | Wayne L. McDaniel |
| author_sort | Wayne L. McDaniel |
| collection | DOAJ |
| description | 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). |
| format | Article |
| id | doaj-art-2cb0f4fc71f140aeb3e10864bd930b07 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1990-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-2cb0f4fc71f140aeb3e10864bd930b072025-08-20T03:55:12ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251990-01-01131132410.1155/S0161171290000023An analogue in certain unique factorization domains of the Euclid-Euler theorem on perfect numbersWayne L. McDaniel0Department of Mathematics and Computer Science, University of Missouri—St. Louis, St. Louis 63121, MO, USAWe 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).http://dx.doi.org/10.1155/S0161171290000023sum of divisorsperfect numbersunique factorization domain. |
| spellingShingle | 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 sum of divisors perfect numbers unique factorization domain. |
| title | An analogue in certain unique factorization domains of the Euclid-Euler theorem on perfect numbers |
| title_full | An analogue in certain unique factorization domains of the Euclid-Euler theorem on perfect numbers |
| title_fullStr | An analogue in certain unique factorization domains of the Euclid-Euler theorem on perfect numbers |
| title_full_unstemmed | An analogue in certain unique factorization domains of the Euclid-Euler theorem on perfect numbers |
| title_short | An analogue in certain unique factorization domains of the Euclid-Euler theorem on perfect numbers |
| title_sort | analogue in certain unique factorization domains of the euclid euler theorem on perfect numbers |
| topic | sum of divisors perfect numbers unique factorization domain. |
| url | http://dx.doi.org/10.1155/S0161171290000023 |
| work_keys_str_mv | AT waynelmcdaniel ananalogueincertainuniquefactorizationdomainsoftheeuclideulertheoremonperfectnumbers AT waynelmcdaniel analogueincertainuniquefactorizationdomainsoftheeuclideulertheoremonperfectnumbers |