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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 a n = 3 + 4 n {\displaystyle a_{n}=3+4n} for 0 ≤ n ≤ 2 {\displaystyle 0\leq n\leq 2} .
Linnik's theorem (1944) concerns the size of the smallest prime in a given arithmetic progression. Linnik proved that the progression a + nd (as n ranges through the positive integers) contains a prime of magnitude at most cd L for absolute constants c and L. Subsequent researchers have reduced L to 5.
In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number, there exist arithmetic progressions of primes with terms.
is an infinite arithmetic progression with modulus 9. In an arithmetic progression, all the numbers have the same remainder when divided by the modulus; in this example, the remainder is 3. Because both the modulus 9 and the remainder 3 are multiples of 3, so is every element in the sequence. Therefore, this progression contains only one prime ...
The sequence of primes numbers contains arithmetic progressions of any length. This result was proven by Ben Green and Terence Tao in 2004 and is now known as the Green–Tao theorem. [3] See also Dirichlet's theorem on arithmetic progressions. As of 2020, the longest known arithmetic progression of primes has length 27: [4]
It is known, based on Dirichlet's theorem on arithmetic progressions, that linear polynomial functions () = + produce infinitely many primes as long as a and b are relatively prime (though no such function will assume prime values for all values of n).
Fix a prime p and note that the (positive, in the Golomb space case) integers are a union of finitely many residue classes modulo p. Each residue class is an arithmetic progression, and thus clopen. Consider the multiples of each prime.
Prime number theorem for arithmetic progressions [ edit ] Let π d , a ( x ) denote the number of primes in the arithmetic progression a , a + d , a + 2 d , a + 3 d , ... that are less than x .