Dickson's conjecture

Dickson's conjecture: In number theory, a branch of mathematics, Dickson's conjecture is the conjecture stated by Dickson (1904) that for a finite set of linear forms $a 1 + nb 1$ , $a 2 + nb 2$ , ..., $a k + nb k$ with $b i \geq 1$ , there are infinitely many positive integers $n$ for which they are all prime, unless there is a congruence condition preventing this (Ribenboim 1996, 6.I). The case k = 1 is Dirichlet's theorem.

Two special cases are well known conjectures: there are infinitely many twin primes (n and n + 2 are primes), and there are infinitely many Sophie Germain primes (n and 2n + 1 are primes).

Dickson's conjecture is further extended by Schinzel's hypothesis H.

See also

Prime triplet

Green–Tao theorem

References

Dickson, L. E. (1904), "A new extension of Dirichlet's theorem on prime numbers", Messenger of mathematics 33: 155–161, http://books.google.com/books?id=i8MKAAAAIAAJ&pg=PA155

Ribenboim, Paulo (1996), The new book of prime number records, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94457-9, MR 1377060, http://books.google.com/books?id=72eg8bFw40kC

Categories:
Conjectures
Conjectures about prime numbers
Prime numbers

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Dickson's conjecture

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