Other Demostrative Perspective of How to See Dirichlet’s Theorem
José William Porras Ferreira *
Escuela Naval de Cadetes, Colombian Naval Academy, Cartagena, Colombia.
Willian de Jesus Caballero Guardo
Escuela Naval de Cadetes, Colombian Naval Academy, Cartagena, Colombia.
*Author to whom correspondence should be addressed.
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.
Keywords: Prime theorem, fundamental theorem of arithmetic, Dirichlet’s theorem, reductio ad absurdum.