An extension of the Euclid-Euler theorem to certain α-perfect numbers
Autor(a) principal: | |
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Data de Publicação: | 2022 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10773/35173 |
Resumo: | In a posthumously published work, Euler proved that all even perfect numbers are of the form 2^(p-1)(2^p−1), where 2^p−1 is a prime number. In this article, we extend Euler’s method for certain α-perfect numbers for which Euler’s result can be generalized. In particular, we use Euler’s method to prove that if N is a 3-perfect number divisible by 6; then either 2 || N or 3 || N. As well, we prove that if N is a 5/2-perfect number divisible by 5, then 2^4 || N, 5^2 || N, and 31^2| N. Finally, for p ∈ {17, 257, 65537}, we prove that there are no 2p/(p−1)-perfect numbers divisible by p. |
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An extension of the Euclid-Euler theorem to certain α-perfect numbersEuclid-Euler theoremα-perfect numberFermat numberAbundancy indexAbundancy outlawIn a posthumously published work, Euler proved that all even perfect numbers are of the form 2^(p-1)(2^p−1), where 2^p−1 is a prime number. In this article, we extend Euler’s method for certain α-perfect numbers for which Euler’s result can be generalized. In particular, we use Euler’s method to prove that if N is a 3-perfect number divisible by 6; then either 2 || N or 3 || N. As well, we prove that if N is a 5/2-perfect number divisible by 5, then 2^4 || N, 5^2 || N, and 31^2| N. Finally, for p ∈ {17, 257, 65537}, we prove that there are no 2p/(p−1)-perfect numbers divisible by p.University of Waterloo2022-11-11T15:22:05Z2022-10-14T00:00:00Z2022-10-14info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10773/35173eng1530-7638Almeida, Paulo J.Cardoso, Gabrielinfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2024-02-22T12:07:32Zoai:ria.ua.pt:10773/35173Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T03:06:11.045411Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
An extension of the Euclid-Euler theorem to certain α-perfect numbers |
title |
An extension of the Euclid-Euler theorem to certain α-perfect numbers |
spellingShingle |
An extension of the Euclid-Euler theorem to certain α-perfect numbers Almeida, Paulo J. Euclid-Euler theorem α-perfect number Fermat number Abundancy index Abundancy outlaw |
title_short |
An extension of the Euclid-Euler theorem to certain α-perfect numbers |
title_full |
An extension of the Euclid-Euler theorem to certain α-perfect numbers |
title_fullStr |
An extension of the Euclid-Euler theorem to certain α-perfect numbers |
title_full_unstemmed |
An extension of the Euclid-Euler theorem to certain α-perfect numbers |
title_sort |
An extension of the Euclid-Euler theorem to certain α-perfect numbers |
author |
Almeida, Paulo J. |
author_facet |
Almeida, Paulo J. Cardoso, Gabriel |
author_role |
author |
author2 |
Cardoso, Gabriel |
author2_role |
author |
dc.contributor.author.fl_str_mv |
Almeida, Paulo J. Cardoso, Gabriel |
dc.subject.por.fl_str_mv |
Euclid-Euler theorem α-perfect number Fermat number Abundancy index Abundancy outlaw |
topic |
Euclid-Euler theorem α-perfect number Fermat number Abundancy index Abundancy outlaw |
description |
In a posthumously published work, Euler proved that all even perfect numbers are of the form 2^(p-1)(2^p−1), where 2^p−1 is a prime number. In this article, we extend Euler’s method for certain α-perfect numbers for which Euler’s result can be generalized. In particular, we use Euler’s method to prove that if N is a 3-perfect number divisible by 6; then either 2 || N or 3 || N. As well, we prove that if N is a 5/2-perfect number divisible by 5, then 2^4 || N, 5^2 || N, and 31^2| N. Finally, for p ∈ {17, 257, 65537}, we prove that there are no 2p/(p−1)-perfect numbers divisible by p. |
publishDate |
2022 |
dc.date.none.fl_str_mv |
2022-11-11T15:22:05Z 2022-10-14T00:00:00Z 2022-10-14 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10773/35173 |
url |
http://hdl.handle.net/10773/35173 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
1530-7638 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
University of Waterloo |
publisher.none.fl_str_mv |
University of Waterloo |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
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Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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1799137716603977728 |