Characterizations of Midy's Property
ABSTRACT: In 1836 E. Midy published in France an article where he showed that if p is a prime number, such that the smallest repeating sequence of digits in the decimal expansion of 1 p has an even length, when this sequence is broken into two halves of equal length if these parts are added then the...
- Autores:
-
García Pulgarín, Gilberto
Giraldo Salazar, Hernán Alonso
- Tipo de recurso:
- Article of investigation
- Fecha de publicación:
- 2009
- Institución:
- Universidad de Antioquia
- Repositorio:
- Repositorio UdeA
- Idioma:
- eng
- OAI Identifier:
- oai:bibliotecadigital.udea.edu.co:10495/23653
- Acceso en línea:
- http://hdl.handle.net/10495/23653
- Palabra clave:
- Teorema de Midy
Teoría de los números
Numbers, Theory of
- Rights
- openAccess
- License
- http://creativecommons.org/licenses/by/2.5/co/
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Characterizations of Midy's Property |
| title |
Characterizations of Midy's Property |
| spellingShingle |
Characterizations of Midy's Property Teorema de Midy Teoría de los números Numbers, Theory of |
| title_short |
Characterizations of Midy's Property |
| title_full |
Characterizations of Midy's Property |
| title_fullStr |
Characterizations of Midy's Property |
| title_full_unstemmed |
Characterizations of Midy's Property |
| title_sort |
Characterizations of Midy's Property |
| dc.creator.fl_str_mv |
García Pulgarín, Gilberto Giraldo Salazar, Hernán Alonso |
| dc.contributor.author.none.fl_str_mv |
García Pulgarín, Gilberto Giraldo Salazar, Hernán Alonso |
| dc.subject.lemb.none.fl_str_mv |
Teorema de Midy Teoría de los números Numbers, Theory of |
| topic |
Teorema de Midy Teoría de los números Numbers, Theory of |
| description |
ABSTRACT: In 1836 E. Midy published in France an article where he showed that if p is a prime number, such that the smallest repeating sequence of digits in the decimal expansion of 1 p has an even length, when this sequence is broken into two halves of equal length if these parts are added then the result is a string of 9s. Later, J. Lewittes and H. W. Martin generalized this statement when the length of the smallest repeating sequence of digits is e = kd and the sequence is broken into d blocks of equal length and the expansion is over any number base; that fact was named Midy’s property. We will give necessary and sufficient conditions (that are easy to check) for the integer N to satisfy Midy’s property. |
| publishDate |
2009 |
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2009 |
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2021-10-29T22:33:25Z |
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2021-10-29T22:33:25Z |
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http://purl.org/coar/version/c_970fb48d4fbd8a85 |
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http://purl.org/coar/resource_type/c_2df8fbb1 |
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https://purl.org/redcol/resource_type/ART |
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Artículo de investigación |
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http://hdl.handle.net/10495/23653 |
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1553-1732 |
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http://hdl.handle.net/10495/23653 |
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1553-1732 |
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eng |
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eng |
| dc.relation.ispartofjournalabbrev.spa.fl_str_mv |
Integers |
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info:eu-repo/semantics/openAccess |
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http://creativecommons.org/licenses/by/2.5/co/ |
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openAccess |
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http://creativecommons.org/licenses/by/2.5/co/ http://purl.org/coar/access_right/c_abf2 https://creativecommons.org/licenses/by/4.0/ |
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Colgate University, Charles University, and DIMATIA |
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Álgebra Teoría de Números y Aplicaciones: ERM |
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Estados Unidos |
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Universidad de Antioquia |
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andres.perez@udea.edu.co |
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1805390303273680896 |
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García Pulgarín, GilbertoGiraldo Salazar, Hernán Alonso2021-10-29T22:33:25Z2021-10-29T22:33:25Z2009http://hdl.handle.net/10495/236531553-1732ABSTRACT: In 1836 E. Midy published in France an article where he showed that if p is a prime number, such that the smallest repeating sequence of digits in the decimal expansion of 1 p has an even length, when this sequence is broken into two halves of equal length if these parts are added then the result is a string of 9s. Later, J. Lewittes and H. W. Martin generalized this statement when the length of the smallest repeating sequence of digits is e = kd and the sequence is broken into d blocks of equal length and the expansion is over any number base; that fact was named Midy’s property. We will give necessary and sufficient conditions (that are easy to check) for the integer N to satisfy Midy’s property.COL00172177application/pdfengColgate University, Charles University, and DIMATIAÁlgebra Teoría de Números y Aplicaciones: ERMEstados Unidosinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articlehttp://purl.org/coar/resource_type/c_2df8fbb1https://purl.org/redcol/resource_type/ARTArtículo de investigaciónhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by/2.5/co/http://purl.org/coar/access_right/c_abf2https://creativecommons.org/licenses/by/4.0/Characterizations of Midy's PropertyTeorema de MidyTeoría de los númerosNumbers, Theory ofIntegersIntegers1911979ORIGINALGarcíaGilberto_2009_ CharacterizationsMidyProperty.pdfGarcíaGilberto_2009_ CharacterizationsMidyProperty.pdfArtículo de investigaciónapplication/pdf389241http://bibliotecadigital.udea.edu.co/bitstream/10495/23653/1/Garc%c3%adaGilberto_2009_%20CharacterizationsMidyProperty.pdf11ec3331bc313778d9f600a874cebe6eMD51LICENSElicense.txtlicense.txttext/plain; charset=utf-81748http://bibliotecadigital.udea.edu.co/bitstream/10495/23653/3/license.txt8a4605be74aa9ea9d79846c1fba20a33MD53CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8927http://bibliotecadigital.udea.edu.co/bitstream/10495/23653/2/license_rdf1646d1f6b96dbbbc38035efc9239ac9cMD5210495/23653oai:bibliotecadigital.udea.edu.co:10495/236532021-10-29 17:33:26.6Repositorio Institucional Universidad de Antioquiaandres.perez@udea.edu.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 |