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) < bL 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)