The generalized Fermat conjecture
If a, b, c are non-zero integers, we considerer the following problem: for which values of n the line ax + by + cz = 0 may be tangent to the curve x n + y n = z n ? We give a partial solution: if n = 5 or if n-1 is a prime a number, then the answer is the line cannot be tangent to the curve. This pr...
- Autores:
-
García-Máynez, Adalberto
Gary, Margarita
Pimienta Acosta, Adolfo
- Tipo de recurso:
- Article of journal
- Fecha de publicación:
- 2018
- Institución:
- Corporación Universidad de la Costa
- Repositorio:
- REDICUC - Repositorio CUC
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.cuc.edu.co:11323/3331
- Acceso en línea:
- https://hdl.handle.net/11323/3331
https://repositorio.cuc.edu.co/
- Palabra clave:
- Chebyshev polynomials
Fermat curve
Tangent
Polinomios de Chebyshev
Curva de Fermat
Tangente
- Rights
- openAccess
- License
- Attribution-NonCommercial-ShareAlike 4.0 International
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| dc.title.spa.fl_str_mv |
The generalized Fermat conjecture |
| dc.title.translated.spa.fl_str_mv |
La conjetura generalizada de Fermat |
| title |
The generalized Fermat conjecture |
| spellingShingle |
The generalized Fermat conjecture Chebyshev polynomials Fermat curve Tangent Polinomios de Chebyshev Curva de Fermat Tangente |
| title_short |
The generalized Fermat conjecture |
| title_full |
The generalized Fermat conjecture |
| title_fullStr |
The generalized Fermat conjecture |
| title_full_unstemmed |
The generalized Fermat conjecture |
| title_sort |
The generalized Fermat conjecture |
| dc.creator.fl_str_mv |
García-Máynez, Adalberto Gary, Margarita Pimienta Acosta, Adolfo |
| dc.contributor.author.spa.fl_str_mv |
García-Máynez, Adalberto Gary, Margarita Pimienta Acosta, Adolfo |
| dc.subject.spa.fl_str_mv |
Chebyshev polynomials Fermat curve Tangent Polinomios de Chebyshev Curva de Fermat Tangente |
| topic |
Chebyshev polynomials Fermat curve Tangent Polinomios de Chebyshev Curva de Fermat Tangente |
| description |
If a, b, c are non-zero integers, we considerer the following problem: for which values of n the line ax + by + cz = 0 may be tangent to the curve x n + y n = z n ? We give a partial solution: if n = 5 or if n-1 is a prime a number, then the answer is the line cannot be tangent to the curve. This problem is strongly related to Fermat' s Last Theorem. |
| publishDate |
2018 |
| dc.date.issued.none.fl_str_mv |
2018-05-05 |
| dc.date.accessioned.none.fl_str_mv |
2019-05-15T13:47:03Z |
| dc.date.available.none.fl_str_mv |
2019-05-15T13:47:03Z |
| dc.type.spa.fl_str_mv |
Artículo de revista |
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http://purl.org/coar/resource_type/c_2df8fbb1 |
| dc.type.coar.spa.fl_str_mv |
http://purl.org/coar/resource_type/c_6501 |
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Text |
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info:eu-repo/semantics/article |
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http://purl.org/redcol/resource_type/ART |
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info:eu-repo/semantics/acceptedVersion |
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http://purl.org/coar/resource_type/c_6501 |
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acceptedVersion |
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01399918 |
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https://hdl.handle.net/11323/3331 |
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Corporación Universidad de la Costa |
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REDICUC - Repositorio CUC |
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https://repositorio.cuc.edu.co/ |
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01399918 Corporación Universidad de la Costa REDICUC - Repositorio CUC |
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https://hdl.handle.net/11323/3331 https://repositorio.cuc.edu.co/ |
| dc.language.iso.none.fl_str_mv |
eng |
| language |
eng |
| dc.relation.references.spa.fl_str_mv |
Fine, B.—Rosenberger, G.: Classification of all generating pairs of two generator Fuchsian groups. In: London Math. Soc. Lecture Note Ser. 211, 1995, pp. 205–232. Garling, D. J. H.: A Course in Galois Theory, Cambridge University Press, 1986. Lang, S.: Cyclotomic Fields I and II. Graduate Texts in Math. 121, Springer-Verlag, New York, 1990. Silverman, J. H.: Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Math. 151, Springer-Verlag, New York, 1994. Washington, L.: Introduction to Cyclotomic Fields. Graduate Texts in Math., Springer-Verlag, New York, 1996. Wiles, A.: Modular elliptic curves and Fermat’s Last Theorem, Ann. Math. 141 (1995), 443–55 |
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Attribution-NonCommercial-ShareAlike 4.0 International |
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http://creativecommons.org/licenses/by-nc-sa/4.0/ |
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Universidad de la Costa |
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Corporación Universidad de la Costa |
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García-Máynez, AdalbertoGary, MargaritaPimienta Acosta, Adolfo2019-05-15T13:47:03Z2019-05-15T13:47:03Z2018-05-0501399918https://hdl.handle.net/11323/3331Corporación Universidad de la CostaREDICUC - Repositorio CUChttps://repositorio.cuc.edu.co/If a, b, c are non-zero integers, we considerer the following problem: for which values of n the line ax + by + cz = 0 may be tangent to the curve x n + y n = z n ? We give a partial solution: if n = 5 or if n-1 is a prime a number, then the answer is the line cannot be tangent to the curve. This problem is strongly related to Fermat' s Last Theorem.Si a, b, c son enteros distintos de cero, consideramos el siguiente problema: ¿para qué valores de n la línea ax + by + cz = 0 pueden ser tangentes a la curva x n + y n = z n? Damos una solución parcial: si n = 5 o si n-1 es un número primo, entonces la respuesta es que la línea no puede ser tangente a la curva. Este problema está fuertemente relacionado con el último teorema de Fermat.García-Máynez, Adalberto-ff8e08c1-a89b-40c2-bf8f-1f7527431018-0Gary, Margarita-d15f7926-7be7-497c-8669-aa91477e8e8d-0Pimienta Acosta, Adolfo-1adedaf2-674d-4a2a-8dc9-208ac359a128-0engUniversidad de la CostaAttribution-NonCommercial-ShareAlike 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-sa/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Chebyshev polynomialsFermat curveTangentPolinomios de ChebyshevCurva de FermatTangenteThe generalized Fermat conjectureLa conjetura generalizada de FermatArtículo de revistahttp://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1Textinfo:eu-repo/semantics/articlehttp://purl.org/redcol/resource_type/ARTinfo:eu-repo/semantics/acceptedVersionFine, B.—Rosenberger, G.: Classification of all generating pairs of two generator Fuchsian groups. In: London Math. Soc. Lecture Note Ser. 211, 1995, pp. 205–232. Garling, D. J. H.: A Course in Galois Theory, Cambridge University Press, 1986. Lang, S.: Cyclotomic Fields I and II. Graduate Texts in Math. 121, Springer-Verlag, New York, 1990. Silverman, J. H.: Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Math. 151, Springer-Verlag, New York, 1994. Washington, L.: Introduction to Cyclotomic Fields. Graduate Texts in Math., Springer-Verlag, New York, 1996. Wiles, A.: Modular elliptic curves and Fermat’s Last Theorem, Ann. Math. 141 (1995), 443–55PublicationORIGINALTHE GENERALIZED FERMAT CONJECTURE.pdfTHE GENERALIZED FERMAT CONJECTURE.pdfapplication/pdf5570https://repositorio.cuc.edu.co/bitstreams/b307d670-9df3-40cb-937c-ddf8ee8c9ba0/downloadba5cb33e20cf4b625078c80485b802b6MD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-81031https://repositorio.cuc.edu.co/bitstreams/55560431-89e9-46a6-acd5-0a068e3d9f85/download934f4ca17e109e0a05eaeaba504d7ce4MD52LICENSElicense.txtlicense.txttext/plain; charset=utf-81748https://repositorio.cuc.edu.co/bitstreams/596f0368-0c9b-4d15-bc7b-a9e530a7111c/download8a4605be74aa9ea9d79846c1fba20a33MD53THUMBNAILTHE GENERALIZED FERMAT CONJECTURE.pdf.jpgTHE GENERALIZED FERMAT CONJECTURE.pdf.jpgimage/jpeg27985https://repositorio.cuc.edu.co/bitstreams/a7982b82-b61b-4bda-a5dc-78b9fd7e189e/downloadaf75febe1c1e0dbe7b117c0883c3adcbMD55TEXTTHE GENERALIZED FERMAT CONJECTURE.pdf.txtTHE GENERALIZED FERMAT CONJECTURE.pdf.txttext/plain542https://repositorio.cuc.edu.co/bitstreams/a420a117-21ec-4b06-ac37-c74ce41cddc6/download30e78048554f2a5cec53ec724b06347fMD5611323/3331oai:repositorio.cuc.edu.co:11323/33312024-09-17 14:21:39.086http://creativecommons.org/licenses/by-nc-sa/4.0/Attribution-NonCommercial-ShareAlike 4.0 Internationalopen.accesshttps://repositorio.cuc.edu.coRepositorio de la Universidad de la Costa CUCrepdigital@cuc.edu.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 |