Consideraciones acerca de la conjetura de Artin sobre raíces primitivas.
diagramas, tablas
- Autores:
-
Alzate Restrepo, Juan David
- Tipo de recurso:
- Fecha de publicación:
- 2021
- Institución:
- Universidad Nacional de Colombia
- Repositorio:
- Universidad Nacional de Colombia
- Idioma:
- spa
- OAI Identifier:
- oai:repositorio.unal.edu.co:unal/80569
- Palabra clave:
- 510 - Matemáticas
Mathematics
Matemáticas
Conjetura de Artin
Raíz primitiva
Hipótesis de Riemann
Primos seguros
Teorema de Hooley
Teorema de Gupta y Murty
Teorema de Heath-Brown
Artin’s Conjecture
Primitive root
Safe primes
Riemann hypothesis
Hooley’s theorem
Gupta and Murty’s theorem
Heath-Brown’s theorem
- Rights
- openAccess
- License
- Atribución-NoComercial-CompartirIgual 4.0 Internacional
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Consideraciones acerca de la conjetura de Artin sobre raíces primitivas. |
| dc.title.translated.eng.fl_str_mv |
A remark on Artin's primitive root conjecture. |
| title |
Consideraciones acerca de la conjetura de Artin sobre raíces primitivas. |
| spellingShingle |
Consideraciones acerca de la conjetura de Artin sobre raíces primitivas. 510 - Matemáticas Mathematics Matemáticas Conjetura de Artin Raíz primitiva Hipótesis de Riemann Primos seguros Teorema de Hooley Teorema de Gupta y Murty Teorema de Heath-Brown Artin’s Conjecture Primitive root Safe primes Riemann hypothesis Hooley’s theorem Gupta and Murty’s theorem Heath-Brown’s theorem |
| title_short |
Consideraciones acerca de la conjetura de Artin sobre raíces primitivas. |
| title_full |
Consideraciones acerca de la conjetura de Artin sobre raíces primitivas. |
| title_fullStr |
Consideraciones acerca de la conjetura de Artin sobre raíces primitivas. |
| title_full_unstemmed |
Consideraciones acerca de la conjetura de Artin sobre raíces primitivas. |
| title_sort |
Consideraciones acerca de la conjetura de Artin sobre raíces primitivas. |
| dc.creator.fl_str_mv |
Alzate Restrepo, Juan David |
| dc.contributor.advisor.none.fl_str_mv |
Toro Villegas, Margarita María |
| dc.contributor.author.none.fl_str_mv |
Alzate Restrepo, Juan David |
| dc.subject.ddc.spa.fl_str_mv |
510 - Matemáticas |
| topic |
510 - Matemáticas Mathematics Matemáticas Conjetura de Artin Raíz primitiva Hipótesis de Riemann Primos seguros Teorema de Hooley Teorema de Gupta y Murty Teorema de Heath-Brown Artin’s Conjecture Primitive root Safe primes Riemann hypothesis Hooley’s theorem Gupta and Murty’s theorem Heath-Brown’s theorem |
| dc.subject.lemb.eng.fl_str_mv |
Mathematics |
| dc.subject.lemb.spa.fl_str_mv |
Matemáticas |
| dc.subject.proposal.spa.fl_str_mv |
Conjetura de Artin Raíz primitiva Hipótesis de Riemann Primos seguros Teorema de Hooley Teorema de Gupta y Murty Teorema de Heath-Brown |
| dc.subject.proposal.eng.fl_str_mv |
Artin’s Conjecture Primitive root Safe primes Riemann hypothesis Hooley’s theorem Gupta and Murty’s theorem Heath-Brown’s theorem |
| description |
diagramas, tablas |
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2021 |
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2021-10-16T17:35:21Z |
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2021-10-16T17:35:21Z |
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2021-10-11 |
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Trabajo de grado - Maestría |
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info:eu-repo/semantics/masterThesis |
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Universidad Nacional de Colombia |
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Repositorio Institucional Universidad Nacional de Colombia |
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https://repositorio.unal.edu.co/ |
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Universidad Nacional de Colombia Repositorio Institucional Universidad Nacional de Colombia |
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spa |
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spa |
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E. Artin, Collected papers. Reading, Mass. Addison-Wesley (1965). A.R. Booker, Artins conjecture, Turings method, and the riemann hypothesis. Exp. Math. 15(4), (2006) 385-407. E. Bombieri, Le grand crible dans la théorie analytique des nombres, Astérisque 18, Societé Mathematique de France, (1974). N.A. Carella, Corrigendum The Generalized Artin Primitive Root Conjecture, European J. of pure and applied math. Vol. 11, No. 1, (2018), 23-34. H. Cohen, A Course in Computational Algebraic Number Theory (3rd ed). Springer-Verlag, New York (1996). H. Cohen, Number Theory Volume I: Tools and Diophantine Equiations. Springer-Verlag, New York (2007). J.R. Goldman, The Queen of mathematics A historically Motivated Guide to Number Theory. A K Peters, Natick, Massachusetts. (1997). L.J. Goldstein, Density questions in algebraic number theory. Amer. math. Monthly78, (1971) 342-351. R. Gupta, M. Ram Murty. A remark on Artins conjecture, Inventiones Math. 78 (1984) 127-130. R. Gupta, V. Kumar Murty y M. Ram Murty. The euclidean algorithm for S integers, CMS Conference Proceedings, volume 7, (1985) 189-202. S. S. Gupta, Artin s Conjecture: Unconditional Approach and Elliptic Analogue, UWSpace(2008). G.H. Hardy , E.M. Wright, An introduction to the theory of numbers, OUP (2008). D.R. Heath-Brown. Artin s conjecture for primitive roots, Quart. J. Math. Oxford, (2) 37(1986) 27-38. C. Hooley, On Artin s conjecture, J. reine angnew. Math, 226. (1967) 209-220. H. Iwaniech. Primes of type phi(x; y)+A, where phi is a quadratic form. Acta Arith. 21(1972) 203-224. H. Iwaniec, Rosser s sieve, Acta Arith, 36:, (1980). 171-202. H. Iwaniec. A new form of error term in the linear sieve. Acta Arith., 37:, (1980). 307-320. J. C. Lagarias, A. M. Odlyzko, Efective versions of the Chebotarev density theorem,Algebraic Number Fields (A. Frohlich, ed.) New York, Academic Press,(1977), 409-464. S. Lang, On the zeta function of number fields, Invent. Math. 12 (1971), 337-345. D.H. Lehmer, E. Lehmer, Heuristics, anyone?, Studies in Mathematical Analysis and Related Topics, Stanford Univ. Press, Stanford, (1962), 202-210. H.L. Montgomery, R.C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, Cambridge, 2007. P. Moree, Artins Primitive Root Conjecture A Survey. Integers, (2012). P. Moree, Approximation of singular series and automata. manuscripta mathematica 101,(2000) 385-399. M.R. Murty, Artins conjecture for primitive roots. The Mathematical Intelligencer 10, (1988) 59-67. D. Pallab, B. Kumar. An Analogue Of Artins primitive root conjecture. Integers.16 (2016). W. Raji, An Introductory Course in Elementary Number Theory. Saylor Foundation,Washington. (2013). D. Shanks - Review of: R. Baillie, Data on Artins conjecture. Math. Comp.29, 1164-1165 (1975). D. Shanks - Solved and unsolved problems in number theory- Chelsea Publishing Company (1978). P. Stevenhagen, H.W. Lenstra, jr., Chebotarev and his density theorem, Math. Intelligencer 18 (1996), 26-37. R. Wilson, J. Gray-Mathematical Conversations. (p. 113-127) M. Ram Murty-Artin´s Conjecture for primitive roots. Springer, New York, (2001). |
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xii, 91 páginas |
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Universidad Nacional de Colombia |
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Medellín - Ciencias - Maestría en Ciencias - Matemáticas |
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Escuela de matemáticas |
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Facultad de Ciencias |
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Medellín, Colombia |
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Atribución-NoComercial-CompartirIgual 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-sa/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Toro Villegas, Margarita Maríaabaa6c07e2063dce2203518ab8a5075eAlzate Restrepo, Juan David19acca042659fae4ab134e076dcfc1602021-10-16T17:35:21Z2021-10-16T17:35:21Z2021-10-11https://repositorio.unal.edu.co/handle/unal/80569Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/diagramas, tablasEn este trabajo, nos centramos en estudiar la conjetura de Artin sobre raíces primitivas, hablamos sobre los argumentos heurísticos de Artin para plantear su idea, y estudiamos los teoremas más importantes referente a la conjetura hasta la fecha: El teorema de Hooley donde demuestra la conjetura bajo la hipótesis extendida de Riemann, el teorema de Gupta y Murty, que establece incondicionalmente la validez de la conjetura para al menos un a en S, donde S es un conjunto de 13 elementos perteneciente a cierta familia de conjuntos, y el teorema de Heath-Brown, donde mejora este resultado a una familia de conjuntos S de 3 elementos. También realizamos un estudio de familias específicas de primos F, donde pensamos en la conjetura en el contexto particular de esa familia, donde logramos demostrar qué condiciones debe cumplir p en F para que a = 2; 3; 5 sea raíz primitiva. Por otro lado, realizamos cómputos para los primos p = 2qr+1, de donde se verifica una densidad estable de primos para los cuales a es raíz primitiva, para ciertos valores de a; y a su vez también planteamos algunas conjeturas. (Texto tomado de la fuente)In this paper, we focused on studying the Artin’s conjecture on primitive roots, we discussed about the heuristic arguments thought by Artin to propose his idea, and we studied the most important theorems related to the conjecture till the date: Hooley’s theorem in which he proves the conjecture under the assumption of the extended Riemann hypothesis , Gupta and Murty’s theorem, which proves unconditionally that the conjecture holds for at least an ∈ , where is a set of 13 elements, belonging to a particular set family, and Heath-Brown’s theorem, where this result is improved to a particular set family of sets of size 3 . We also studied some specific prime families and thought about the conjecture in the particular context of that family, we discussed about necessary conditions of ∈ in order to = 2, 3, 5 be a primitive root. In other hand, we performed computations for primes of the form = 2 + 1, and according to those computations it’s verified a estable density for primes for which is a primitive root, for some values of de ; also based on the computations we proposed some conjectures.MaestríaMagíster en Ciencias - MatemáticasTeoría de Númerosxii, 91 páginasapplication/pdfspaUniversidad Nacional de ColombiaMedellín - Ciencias - Maestría en Ciencias - MatemáticasEscuela de matemáticasFacultad de CienciasMedellín, ColombiaUniversidad Nacional de Colombia - Sede Medellín510 - MatemáticasMathematicsMatemáticasConjetura de ArtinRaíz primitivaHipótesis de RiemannPrimos segurosTeorema de HooleyTeorema de Gupta y MurtyTeorema de Heath-BrownArtin’s ConjecturePrimitive rootSafe primesRiemann hypothesisHooley’s theoremGupta and Murty’s theoremHeath-Brown’s theoremConsideraciones acerca de la conjetura de Artin sobre raíces primitivas.A remark on Artin's primitive root conjecture.Trabajo de grado - Maestríainfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/acceptedVersionTexthttp://purl.org/redcol/resource_type/TME. Artin, Collected papers. Reading, Mass. Addison-Wesley (1965).A.R. Booker, Artins conjecture, Turings method, and the riemann hypothesis. Exp. Math. 15(4), (2006) 385-407.E. Bombieri, Le grand crible dans la théorie analytique des nombres, Astérisque 18, Societé Mathematique de France, (1974).N.A. Carella, Corrigendum The Generalized Artin Primitive Root Conjecture, European J. of pure and applied math. Vol. 11, No. 1, (2018), 23-34.H. Cohen, A Course in Computational Algebraic Number Theory (3rd ed). Springer-Verlag, New York (1996).H. Cohen, Number Theory Volume I: Tools and Diophantine Equiations. Springer-Verlag, New York (2007).J.R. Goldman, The Queen of mathematics A historically Motivated Guide to Number Theory. A K Peters, Natick, Massachusetts. (1997).L.J. Goldstein, Density questions in algebraic number theory. Amer. math. Monthly78, (1971) 342-351.R. Gupta, M. Ram Murty. A remark on Artins conjecture, Inventiones Math. 78 (1984) 127-130.R. Gupta, V. Kumar Murty y M. Ram Murty. The euclidean algorithm for S integers, CMS Conference Proceedings, volume 7, (1985) 189-202.S. S. Gupta, Artin s Conjecture: Unconditional Approach and Elliptic Analogue, UWSpace(2008).G.H. Hardy , E.M. Wright, An introduction to the theory of numbers, OUP (2008).D.R. Heath-Brown. Artin s conjecture for primitive roots, Quart. J. Math. Oxford, (2) 37(1986) 27-38.C. Hooley, On Artin s conjecture, J. reine angnew. Math, 226. (1967) 209-220.H. Iwaniech. Primes of type phi(x; y)+A, where phi is a quadratic form. Acta Arith. 21(1972) 203-224.H. Iwaniec, Rosser s sieve, Acta Arith, 36:, (1980). 171-202.H. Iwaniec. A new form of error term in the linear sieve. Acta Arith., 37:, (1980). 307-320.J. C. Lagarias, A. M. Odlyzko, Efective versions of the Chebotarev density theorem,Algebraic Number Fields (A. Frohlich, ed.) New York, Academic Press,(1977), 409-464.S. Lang, On the zeta function of number fields, Invent. Math. 12 (1971), 337-345.D.H. Lehmer, E. Lehmer, Heuristics, anyone?, Studies in Mathematical Analysis and Related Topics, Stanford Univ. Press, Stanford, (1962), 202-210.H.L. Montgomery, R.C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, Cambridge, 2007.P. Moree, Artins Primitive Root Conjecture A Survey. Integers, (2012).P. Moree, Approximation of singular series and automata. manuscripta mathematica 101,(2000) 385-399.M.R. Murty, Artins conjecture for primitive roots. The Mathematical Intelligencer 10, (1988) 59-67.D. Pallab, B. Kumar. An Analogue Of Artins primitive root conjecture. Integers.16 (2016).W. Raji, An Introductory Course in Elementary Number Theory. Saylor Foundation,Washington. (2013).D. Shanks - Review of: R. Baillie, Data on Artins conjecture. Math. Comp.29, 1164-1165 (1975).D. Shanks - Solved and unsolved problems in number theory- Chelsea Publishing Company (1978).P. Stevenhagen, H.W. Lenstra, jr., Chebotarev and his density theorem, Math. Intelligencer 18 (1996), 26-37.R. Wilson, J. Gray-Mathematical Conversations. (p. 113-127) M. Ram Murty-Artin´s Conjecture for primitive roots. Springer, New York, (2001).FP44842-013-2018Hermes, código 50652Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación.COLCIENCIASUniversidad Nacional de Colombia-Sede MedellínInvestigadoresLICENSElicense.txtlicense.txttext/plain; charset=utf-83964https://repositorio.unal.edu.co/bitstream/unal/80569/3/license.txtcccfe52f796b7c63423298c2d3365fc6MD53ORIGINAL1152446073.2021.pdf1152446073.2021.pdfTesis de Maestría en Ciencias - Matemáticasapplication/pdf66090556https://repositorio.unal.edu.co/bitstream/unal/80569/4/1152446073.2021.pdfc0d54de60025b8526d4546f8c9a4ff76MD54THUMBNAIL1152446073.2021.pdf.jpg1152446073.2021.pdf.jpgGenerated Thumbnailimage/jpeg4306https://repositorio.unal.edu.co/bitstream/unal/80569/5/1152446073.2021.pdf.jpg7b884498a38ed38dd69d18610e95227aMD55unal/80569oai:repositorio.unal.edu.co:unal/805692023-07-30 23:03:29.344Repositorio Institucional Universidad Nacional de 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