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). 