On Special Cases of Dirichlet’s Theorem on Arithmetic Progressions

Sammanfattning: Dirichlet’s theorem regarding existence of infinitely many primes in progressions on the form a, a + n, a + 2n... when (a,n) = 1 is well known and proved by using Dirichlet series. This thesis will mainly treat the special case when a = 1 without the use of such series. In the first section of the thesis we show existence of an upper bound as a function of n for when the first prime occurs in progressions of this form. The second section contains proofs of the existence of infinitely many primes in progressions when a = 1 and n being 4,6,8 and finally n being an arbitrary integer, using only elementary methods. In the last section we look into some results in algebraic number theory.