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Research Article Open access CC BY 4.0

Other Demostrative Perspective of How to See Dirichlet’s Theorem

José William Porras Ferreira, Willian de Jesus Caballero Guardo

Journal of Scientific Research and Reports · pp. 1–7 · Published 3 Mar 2016

10.9734/JSRR/2016/24470

Abstract

The Dirichlet’s theorem (1837), initially guessed by Gauss, is a result of analytic number theory. Dirichlet, demonstrated that: For any two positive coprime integers and , there are infinite primes of the form a+bn, where n is a non-negative integer ( n = 1, 2,… ). In other words, there are infinite primes which are congruent to mod b. The numbers of the form a+bn is an arithmetic progression. Actually, Dirichlet checks a result somewhat more interesting than the previous claim, since he demonstrated that: Which implies that there are infinite primes, p  a mod b. The proof of the theorem uses the properties of certain Dirichlet L-functions and some results on arithmetic of complex numbers, and it is sufficiently complex that some texts about numbers theory excluded it. Here is a simple proof by reductio ad absurdum which does not require extensive mathematical knowledge.

Prime theorem fundamental theorem of arithmetic Dirichlet’s theorem reductio ad absurdum.

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