The least prime in certain arithmetic progressions
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...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2009 |
| País: | Argentina |
| Institución: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/156292 |
| Acceso en línea: | http://hdl.handle.net/11336/156292 |
| Access Level: | acceso abierto |
| Palabra clave: | PRIME NUMBERS ARITHMETIC PROGRESSIONS https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Sumario: | 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. |
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