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