The least prime in certain arithmetic progressions

Autores
Sabia, Juan Vicente Rafael; Tesauri, Susana
Año de publicación
2009
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Dirichlet’s theorem states that, if a and n are relatively prime integers, there are infinitely many primes in the arithmetic progression n + a, 2n + a, 3n + a,.... However, the known proofs of this general result are not elementary (see [1, 10, 12], for example). Linnik [4, 5] proved that, if 1 ≤ a < n, there are absolute constants c1 and c2 so that the least prime p in such a progression satisfies p ≤ c1nc2 , but his proof is not elementary either. There are several different proofs of Dirichlet’s theorem for the particular case a = 1 (see for example [2, 6, 9, 11]). In [7], moreover, the bound p < n3n for the least prime satisfying p ≡ 1 (mod n) is given. Our aim is to use an elementary argument, which also shows that there are infinitely many primes ≡ 1 (mod n), to prove that the least such prime lies below (3n − 1)/2.
Fil: Sabia, Juan Vicente Rafael. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Tesauri, Susana. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina
Materia
PRIME NUMBERS
ARITHMETIC PROGRESSIONS
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
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spelling The least prime in certain arithmetic progressionsSabia, Juan Vicente RafaelTesauri, SusanaPRIME NUMBERSARITHMETIC PROGRESSIONShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Dirichlet’s theorem states that, if a and n are relatively prime integers, there are infinitely many primes in the arithmetic progression n + a, 2n + a, 3n + a,.... However, the known proofs of this general result are not elementary (see [1, 10, 12], for example). Linnik [4, 5] proved that, if 1 ≤ a < n, there are absolute constants c1 and c2 so that the least prime p in such a progression satisfies p ≤ c1nc2 , but his proof is not elementary either. There are several different proofs of Dirichlet’s theorem for the particular case a = 1 (see for example [2, 6, 9, 11]). In [7], moreover, the bound p < n3n for the least prime satisfying p ≡ 1 (mod n) is given. Our aim is to use an elementary argument, which also shows that there are infinitely many primes ≡ 1 (mod n), to prove that the least such prime lies below (3n − 1)/2.Fil: Sabia, Juan Vicente Rafael. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Tesauri, Susana. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; ArgentinaMathematical Association of America2009-12info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/156292Sabia, Juan Vicente Rafael; Tesauri, Susana; The least prime in certain arithmetic progressions; Mathematical Association of America; The American Mathematical Monthly; 116; 7; 12-2009; 641-6430002-9890CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1080/00029890.2009.11920982info:eu-repo/semantics/altIdentifier/url/https://www.tandfonline.com/doi/abs/10.1080/00029890.2009.11920982info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T09:44:50Zoai:ri.conicet.gov.ar:11336/156292instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-03 09:44:50.413CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv The least prime in certain arithmetic progressions
title The least prime in certain arithmetic progressions
spellingShingle The least prime in certain arithmetic progressions
Sabia, Juan Vicente Rafael
PRIME NUMBERS
ARITHMETIC PROGRESSIONS
title_short The least prime in certain arithmetic progressions
title_full The least prime in certain arithmetic progressions
title_fullStr The least prime in certain arithmetic progressions
title_full_unstemmed The least prime in certain arithmetic progressions
title_sort The least prime in certain arithmetic progressions
dc.creator.none.fl_str_mv Sabia, Juan Vicente Rafael
Tesauri, Susana
author Sabia, Juan Vicente Rafael
author_facet Sabia, Juan Vicente Rafael
Tesauri, Susana
author_role author
author2 Tesauri, Susana
author2_role author
dc.subject.none.fl_str_mv PRIME NUMBERS
ARITHMETIC PROGRESSIONS
topic PRIME NUMBERS
ARITHMETIC PROGRESSIONS
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv Dirichlet’s theorem states that, if a and n are relatively prime integers, there are infinitely many primes in the arithmetic progression n + a, 2n + a, 3n + a,.... However, the known proofs of this general result are not elementary (see [1, 10, 12], for example). Linnik [4, 5] proved that, if 1 ≤ a < n, there are absolute constants c1 and c2 so that the least prime p in such a progression satisfies p ≤ c1nc2 , but his proof is not elementary either. There are several different proofs of Dirichlet’s theorem for the particular case a = 1 (see for example [2, 6, 9, 11]). In [7], moreover, the bound p < n3n for the least prime satisfying p ≡ 1 (mod n) is given. Our aim is to use an elementary argument, which also shows that there are infinitely many primes ≡ 1 (mod n), to prove that the least such prime lies below (3n − 1)/2.
Fil: Sabia, Juan Vicente Rafael. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Tesauri, Susana. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina
description Dirichlet’s theorem states that, if a and n are relatively prime integers, there are infinitely many primes in the arithmetic progression n + a, 2n + a, 3n + a,.... However, the known proofs of this general result are not elementary (see [1, 10, 12], for example). Linnik [4, 5] proved that, if 1 ≤ a < n, there are absolute constants c1 and c2 so that the least prime p in such a progression satisfies p ≤ c1nc2 , but his proof is not elementary either. There are several different proofs of Dirichlet’s theorem for the particular case a = 1 (see for example [2, 6, 9, 11]). In [7], moreover, the bound p < n3n for the least prime satisfying p ≡ 1 (mod n) is given. Our aim is to use an elementary argument, which also shows that there are infinitely many primes ≡ 1 (mod n), to prove that the least such prime lies below (3n − 1)/2.
publishDate 2009
dc.date.none.fl_str_mv 2009-12
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/156292
Sabia, Juan Vicente Rafael; Tesauri, Susana; The least prime in certain arithmetic progressions; Mathematical Association of America; The American Mathematical Monthly; 116; 7; 12-2009; 641-643
0002-9890
CONICET Digital
CONICET
url http://hdl.handle.net/11336/156292
identifier_str_mv Sabia, Juan Vicente Rafael; Tesauri, Susana; The least prime in certain arithmetic progressions; Mathematical Association of America; The American Mathematical Monthly; 116; 7; 12-2009; 641-643
0002-9890
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1080/00029890.2009.11920982
info:eu-repo/semantics/altIdentifier/url/https://www.tandfonline.com/doi/abs/10.1080/00029890.2009.11920982
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Mathematical Association of America
publisher.none.fl_str_mv Mathematical Association of America
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 13.13397

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