Given an arithmetic progression of terms primes
if relatively prime , i.e.,
Dirichlet proved this theorem using Dirichlet L-series , but the proof is challenging enough that, in their classic text on number
theory , the usually explicit Hardy and Wright (1979) report "this theorem
is too difficult for insertion in this book."
See also Bouniakowsky Conjecture ,
k -Tuple Conjecture,
Modular
Prime Counting Function ,
Prime Arithmetic
Progression ,
Relatively Prime ,
Sierpiński's
Prime Sequence Theorem
Explore with Wolfram|Alpha
References Courant, R. and Robbins, H. "Primes in Arithmetical Progressions." §1.2b in Supplement to Ch. 1 in What
Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Derbyshire, J.
Prime
Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Dirichlet, L. "Beweis des
Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz
ganze Zahlen ohne gemeinschaftlichen Factor sing, unendlich viele Primzahlen erhält."
Abhandlungen der Königlich Preussischen Akademie der Wissenschaften ,
pp. 45-81, 1837. Hardy, G. H. and Wright, E. M. An
Introduction to the Theory of Numbers, 5th ed. Havil, J. Gamma:
Exploring Euler's Constant. Landau, E. Vorlesungen
über Zahlentheorie, Vol. 1. Landau, E. Handbuch
der Lehre von der Verteilung der Primzahlen, 3rd ed. Shanks, D. Solved
and Unsolved Problems in Number Theory, 4th ed. Referenced on Wolfram|Alpha Dirichlet's Theorem
Cite this as:
Weisstein, Eric W. "Dirichlet's Theorem."
From MathWorld https://mathworld.wolfram.com/DirichletsTheorem.html
Subject classifications
,
for 
,
2, ..., the series contains an infinite number of primes
if 
and 
are relatively prime, i.e., 
. This result had been conjectured by Gauss (Derbyshire
2004, p. 96), but was first proved by Dirichlet (1837).
