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 a₁ + nb₁, a₂ + nb₂, ..., a_k + nb_k with b_i ≥ 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
• Ribenboim, Paulo (1996), The new book of prime number records, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94457-9, MR1377060