Dickson's conjecture
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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 a1 + b1n, a2 + b2n, ..., ak + bkn with bi ≥ 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 other special cases are well known conjectures: there are infinitely many twin primes (n and 2 + n are primes), and there are infinitely many Sophie Germain primes (n and 1 + 2n are primes).
Dickson's conjecture is further extended by Schinzel's hypothesis H.
See also
- Prime triplet
- Green–Tao theorem
- Prime constellation
References
- Dickson, L. E. (1904), "A new extension of Dirichlet's theorem on prime numbers", Messenger of mathematics 33: 155–161
- Ribenboim, Paulo (1996), The new book of prime number records, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94457-9, MR 1377060
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