Abstract
The arithmetic progression of odd numbers 1, 3, 5, ... , 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
If 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.
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© 1976 Springer Science+Business Media New York
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Apostol, T.M. (1976). Dirichlet’s Theorem on Primes in Arithmetical Progressions. In: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-28579-4_8
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DOI: https://doi.org/10.1007/978-3-662-28579-4_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-90163-1
Online ISBN: 978-3-662-28579-4
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