# Linnik's Constant

Dirichlet's Theorem on Primes in Arithmetic Progressions states
that if *a* and *b* are relatively prime positive
integers, then the arithmetic progression

a,a+b,a+2b,a+3b, ...,a+nb, ...

contains infinitely many primes. A natural question
to ask is "by when must the first such prime (let's
call it p(*a*,*b*)) occur?" In 1944 Linnik proved that
there exists a constant *L* such that
p(*a*,*b*) < *b ^{L}* for all

*a*and for all sufficiently large

*b*. It is known

*L*is less than 5.5, and that we can take

*L*=2 for almost all integers

*b*. See the web page linked below for much more information.

**Related pages** (outside of this work)

- Linnik's Constant from MathWorld

Printed from the PrimePages <primes.utm.edu> © Chris Caldwell.