Dickson's conjecture 📖 Wikipedia

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

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

Given ${\displaystyle n}$ polynomials with positive degrees and integer coefficients (⁠${\displaystyle n}$⁠ can be any natural number) that each satisfy all three conditions in the Bunyakovsky conjecture, and for any prime ${\displaystyle p}$ there is an integer ${\displaystyle x}$ such that the values of all ${\displaystyle n}$ polynomials at ${\displaystyle x}$ are not divisible by ⁠${\displaystyle p}$⁠, then there are infinitely many positive integers ${\displaystyle x}$ such that all values of these ${\displaystyle n}$ polynomials at ${\displaystyle x}$ are prime. For example, if the conjecture is true then there are infinitely many positive integers ${\displaystyle x}$ such that ⁠${\displaystyle x^{2}+1}$⁠, ⁠${\displaystyle 3x-1}$⁠, and ${\displaystyle x^{2}+x+41}$ 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.

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

• 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