Other Demostrative Perspective of How to See Dirichlet’s Theorem

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 ܽ ܾ݊, where ݊ is a non-negative integer ( ݊ ൌ 1, 2, … ). In other words, there are infinite primes which are congruent to ܽ mod b. The numbers of the form ܽ ܾ݊ 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, ‫‬ ≡ ܽ ‫݀݉‬ ܾ.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.