Two theorems and five conjectures about forms with infinitely many primes

Ernest Davis
September 12, 2022.

Following up on a Facebook post by Scott Aaronson about Greg Kuperberg's bathroom tiles, and a discussion with Joshua Zelinsky about what happens when those are used as the starting point for the Game of Life, I looked into generalizations of the twin prime conjecture and related results. I found that there are quite a lot of those.

Dirichlet's theorem on arithmetic progressions (1837). In any arithmetic sequence a+bn where a and b are relatively prime integers, b ≥ 1, there are infinitely many primes. (Moreover, for a fixed value of b, the different possible values of a all get their fair share of the primes, asymptotically.)

Polignac's conjecture (1849): For any positive even n, there are infinitely many prime gaps of size n. I asked on mathoverflow and apparently, even the weaker (apparently unnamed) conjectures "for any positive even n, there exists at least one prime gap of size n" and the even weaker "for any positive even n, there exist two primes separated by n" have not been proved.

Bunyakovsky's conjecture (1857). If f(x) is a polynomial with integer coefficients satisfying the conditions that
1. The leading coefficient is positive.
2. The polynomial is irreducible.
3. There is no value m that divides all the values f(1), f(2) ...
then f(x) has infinitely many prime values. For polynomials of degree 1, this is Dirichlet's theorem. It has not been proved for any polynomial of degree > 1.

Dickson's conjecture (1904). "For any finite set of linear forms a1+b1*n ... ak+bk*n" there are infinitely many n where all of these are prime, unless there is a congruence condition preventing this" (e.g. there are not infinitely many n such that n, n+2, and n+4 are all prime, because one of those must be divisible by 3). The special case where k=1 is again Dirichlet's theorem.

Generalized Bunyavkovsky's conjecture (also called generalized Dickson's conjecture): Given k polynomials f1 ... fk each of which satisfies the three conditions of Bunyavkovsky's conjecture and additionally
4: For any prime p there is an n such that none of f1(n) ... fk(n) is divisible by p
then there are infinitely many values of n such that all of f1(n) ... fk(n) are prime.

Schinzel's hypothesis H (1958): For any finite collection {f1 ... fk} of non-constant, irreducible polynomials with integer coefficients one of the following holds
1. There is an integer m such that, for all n, m divides at least one of f1(n) .... fk(n).
2. There are infinitely many n such that all of f1(n) ... fk(n) are prime.

More or less, over these categories of functions, either there is a simple proof of a particular form that only there are only finitely many values of n that make all the fi(n) prime or there are indeed infinitely many values that make all the fi(n) prime.

For some reason, these make me very happy. If I were starting out in life as a mathematician, I would (no doubt foolishly) be tempted to work in this direction.

Dirichlet's theorem and Yitang Zhang theorem on prime gaps and its improvements by Polymath are pretty much the only theorems of this flavor that have been proved.

It also made me happy to learn, in the course of this, that Peter Dirichlet married Felix Mendelssohn's youngest sister Rebecka.