Introduction to Analytic Number Theory pp 146-156 | Cite as

# Dirichlet’s Theorem on Primes in Arithmetical Progressions

Chapter

## Abstract

The arithmetic progression of odd numbers 1, 3, 5, ... , If

*2n*+ 1,... contains infinitely many primes. It is natural to ask whether other arithmetic progressions have this property. An arithmetic progression with first term*h*and common difference*k*consists of all numbers of the form$$kn = h,n = 0,1,2,...,$$

(1)

*h*and*k*have a common factor*d*, each term of the progression is divisible by*d*and there can be no more than one prime in the progression if*d >*1. In other words, a necessary condition for the existence of infinitely many primes in the arithmetic progression (1) is that (*h, k) —*1. Dirichlet was the first to prove that this condition is also sufficient. That is, if (*h, k*) = 1 the arithmetic progression (1) contains infinitely many primes. This result, now known as*Dirichlet’s theorem*, will be proved in this chapter.## Preview

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© Springer Science+Business Media New York 1976