Dickson's conjecture

In number theory, Dickson's conjecture is the statement that for a finite set of linear forms $a_{1}+b_{1}n,a_{2}+b_{2}n,\dots ,a_{k}+b_{k}n$ with each ⁠$b_{i}\geq 1$⁠, there are infinitely many positive integers $n$ for which they are all prime, unless there is a congruence condition preventing this. The conjecture is named after Leonard Dickson, who first proposed it in 1904.

The case $k=1$ is Dirichlet's theorem. Two other special cases are well-known conjectures: that there are infinitely many twin primes (⁠$n$⁠ and $n+2$ are primes), and that there are infinitely many Sophie Germain primes (⁠$n$⁠ and $2n+1$ are primes).

Generalized Dickson's conjecture

Main article: Schinzel's hypothesis H

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 ⁠$x^{2}+1$⁠, ⁠$3x-1$⁠, and $x^{2}+x+41$ are all prime. When all the polynomials have degree 1, this is the original Dickson's conjecture. This generalization is equivalent to the generalized Bunyakovsky conjecture and Schinzel's hypothesis H.

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" (PDF), Messenger of Mathematics, 33: 155–161
• Ribenboim, Paulo (1996) [1988], The New Book of Prime Number Records (3rd ed.), Springer New York, ISBN 978-0-387-94457-9, MR 1377060