Linnik's Constant
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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.

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