Given an arithmetic progression of terms ,
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).
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. Oxford,
England: Oxford University Press, pp. 26-27, 1996. Derbyshire, J.
Prime
Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics.
New York: Penguin, pp. 95-97, 2004. 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. Oxford, England: Clarendon
Press, pp. 13-14, 1979. Havil, J. Gamma:
Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 186,
2003. Landau, E. Vorlesungen
über Zahlentheorie, Vol. 1. New York: Chelsea, pp. 79-96,
1970. Landau, E. Handbuch
der Lehre von der Verteilung der Primzahlen, 3rd ed. New York: Chelsea, pp. 422-446,
1974. Shanks, D. Solved
and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 22-23,
1993. Referenced on Wolfram|Alpha Dirichlet's Theorem
Cite this as:
Weisstein, Eric W. "Dirichlet's Theorem."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/DirichletsTheorem.html
Subject classifications