In number theory, the phrase primes in arithmetic progression refers to at least three prime numbers that are consecutive terms in an arithmetic progression, for example the primes (3, 7, 11) (it does not matter that 5 is also prime).
There are arbitrarily long, but not infinitely long, sequences of primes in arithmetic progression. Sometimes (not in this article) the term may also be used about primes which belong to a given arithmetic progression but are not necessarily consecutive terms. Dirichlet's theorem on arithmetic progressions states: If a and b are coprime, then the arithmetic progression a·n + b contains infinitely many primes.
For integer k ≥ 3, an AP-k (also called PAP-k) is k primes in arithmetic progression. An AP-k can be written as k primes of the form a·n + b, for fixed integers a (called the common difference) and b, and k consecutive integer values of n. An AP-k is usually expressed with n = 0 to k − 1. This can always be achieved by defining b to be the first prime in the arithmetic progression.
Read more about Primes In Arithmetic Progression: Properties, Largest Known Primes in AP, Consecutive Primes in Arithmetic Progression, See Also
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