Primes in arithmetic progressions

In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by ${\displaystyle a_{n}=3+4n}$ for ${\displaystyle 0\leq n\leq 2}$.

According to the Green–Tao theorem, there exist arbitrarily long sequences of primes in arithmetic progression. Sometimes the phrase may also be used about primes which belong to an arithmetic progression which also contains composite numbers. For example, it can be used about primes in an arithmetic progression of the form ${\displaystyle an+b}$, where a and b are coprime which according to Dirichlet's theorem on arithmetic progressions contains infinitely many primes, along with infinitely many composites.

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