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 + b₁n, a₂ + b₂n, ..., a_k + b_kn 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). 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.

Generalized Dickson's conjecture

Given n polynomials with positive degrees and integer coefficients (n can be any natural number) that each satisfy all three conditions in the Bunyakovsky conjecture, and for any prime p there is an integer x such that the values of all n polynomials at x are not divisible by p, then there are infinitely many positive integers x such that all values of these n polynomials at x are prime. For example, if the conjecture is true then there are infinitely many positive integers x such that x² + 1, 3x - 1, and x² + x + 41 are all prime. When all the polynomials have degree 1, this is the original Dickson's conjecture.

This more general conjecture is the same as the Generalized Bunyakovsky conjecture.

See also

• Prime triplet
• Green–Tao theorem
• First Hardy–Littlewood conjecture
• Prime constellation
• Primes in arithmetic progression

References

• Dickson, L. E. (1904), "A new extension of Dirichlet's theorem on prime numbers", Messenger of mathematics 33: 155–161, https://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, https://books.google.com/books?id=72eg8bFw40kC