Properties
Any given arithmetic progression of primes has a finite length. In 2004, Ben J. Green and Terence Tao settled an old conjecture by proving the Green–Tao theorem: The primes contain arbitrarily long arithmetic progressions. It follows immediately that there are infinitely many AP-k for any k.
If an AP-k does not begin with the prime k, then the common difference is a multiple of the primorial k# = 2·3·5·...·j, where j is the largest prime ≤ k.
- Proof: Let the AP-k be a·n + b for k consecutive values of n. If a prime p does not divide a, then modular arithmetic says that p will divide every p'th term of the arithmetic progression. (From H.J. Weber, Cor.10 in ``Exceptional Prime Number Twins, Triplets and Multiplets," arXiv:1102.3075. See also Theor.2.3 in ``Regularities of Twin, Triplet and Multiplet Prime Numbers," arXiv:1103.0447,Global J.P.A.Math 8(2012), in press.) If the AP is prime for k consecutive values, then a must therefore be divisible by all primes p ≤ k.
This also shows that an AP with common difference a cannot contain more consecutive prime terms than the value of the smallest prime that does not divide a.
If k is prime then an AP-k can begin with k and have a common difference which is only a multiple of (k−1)# instead of k#. (From H. J. Weber, ``Less Regular Exceptional and Repeating Prime Number Multiplets," arXiv:1105.4092, Sect.3.) For example the AP-3 with primes {3, 5, 7} and common difference 2# = 2, or the AP-5 with primes {5, 11, 17, 23, 29} and common difference 4# = 6. It is conjectured that such examples exist for all primes k. As of 2009, the largest prime for which this is confirmed is k = 17, for this AP-17 found by Phil Carmody in 2001:
- 17 + 11387819007325752·13#·n, for n = 0 to 16.
It follows from widely believed conjectures, such as Dickson's conjecture and some variants of the prime k-tuple conjecture, that if p > 2 is the smallest prime not dividing a, then there are infinitely many AP-(p−1) with common difference a. For example, 5 is the smallest prime not dividing 6, so there is expected to be infinitely many AP-4 with common difference 6, which is called a sexy prime quadruplet. When a = 2, p = 3, it is the twin prime conjecture, with an "AP-2" of 2 primes (b, b + 2).
Read more about this topic: Primes In Arithmetic Progression
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