Dickson's conjecture

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. The conjecture is named after Leonard Dickson, who first proposed it in 1903.

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).

Generalized Dickson's conjecture

Given ${\displaystyle n\geq 1}$ polynomials with positive degrees and integer coefficients that each satisfy all three conditions in the Bunyakovsky conjecture, and if 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.

See also

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

References