请输入您要查询的字词:

 

单词 SpecialCaseOfDirichletsTheoremOnPrimesInArithmeticProgressions
释义

special case of Dirichlet’s theorem on primes in arithmetic progressions


The special case of Dirichlet’s theorem for primes in arithmetic progressions for primes congruentMathworldPlanetmath to 1 modulo q where q itself is a prime can be established by the following elegant modification of Euclid’s proof (http://planetmath.org/ProofThatThereAreInfinitelyManyPrimes).

Let f(n)=nq-1n-1=1+n+n2++nq-1. Let n>1 be an integer, and suppose pf(n). Then nq1(modp) which implies by Lagrange’s theorem that either qp-1 or n1(modp). In other words, every prime divisorPlanetmathPlanetmathPlanetmath of f(n) is congruent to 1 modulo q unless n is congruent to 1 modulo that divisorMathworldPlanetmathPlanetmathPlanetmath.

Suppose there are only finitely many primes that are congruent to 1 modulo q. Let P be twice their productPlanetmathPlanetmath. Note that P2(modq). Let p be any prime divisor of f(P). If p1(modq), then pP which contradicts f(P)1(modP). Therefore, by the above P1(modp). Therefore f(P)1+P+P2++Pq-11+1+1++1q(modp). Since q is prime, it follows that p=q.Then P1(modp) implies P1(modq). However, that is inconsistent with our deductionMathworldPlanetmathPlanetmath that P2(modq) above.Therefore the original assumption that there are only finitely many primes congruent to 1 modulo q is false.

References

  • 1 Henryk Iwaniec and Emmanuel Kowalski. Analytic Number TheoryMathworldPlanetmath, volume 53 of AMS ColloquiumPublications. AMS, 2004.
随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 22:01:48