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
a_{1}+b_{1}*n ...
a_{k}+b_{k}*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 f_{1} ... f_{k} 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
f_{1}(n) ... f_{k}(n) is divisible by p

then there are infinitely many values of n
such that all of f_{1}(n) ... f_{k}(n) are prime.

**
Schinzel's hypothesis H** (1958):
For any finite collection {f_{1} ... f_{k}} 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 f_{1}(n) .... f_{k}(n).

2. There are infinitely many n such that all of
f_{1}(n) ... f_{k}(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 f_{i}(n) prime or there are indeed infinitely many values that
make all the f_{i}(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.