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Below are a couple of idle questions that came up one day when I became curious about "matrix factorizations over $\mathbb Z$". Let's start with size $2$: consider the equation $n= ab-cd$ (*), where $n$ is an integer, and $a,b,c,d$ are prime number.

Question 1: does the equation (*) always have solution for any $n$? How about infinitely many solutions? For example, assuming the twin prime conjecture, there are infinitely many solutions when $n=4$, just take $a=c=2$, and $b,d$ primes such that $b-d=2$. In fact, numbers that are product of two primes are called semi-primes, and there are some literature about semi-prime gap. But perhaps modern number theory can handle this more easily?

Question 2: what happens if we add more assumptions? For example, $n$ big enough, the primes are distinct, the size of matrix increases, etc.

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    $\begingroup$ Pretty much all such questions should follow from the generalized twin primes problem. Given any $n\times n$ matrix, fix $n^2-2$ of the entries to be any primes you want, leaving (say) two neighboring variables in the same row. It should be easy to arrange that the determinant of the resulting matrix will is a linear polynomial of the form $Ax-By$ in the two variables $x$ and $y$, where $A$ and $B$ are positive coprime integers. Then conjecturally, that linear polynomial evaluated at primes $x$ and $y$ should represent every integer $n$ of the correct parity infinitely often. $\endgroup$
    – Greg Martin
    Commented Dec 23, 2017 at 18:26
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    $\begingroup$ @Greg, shouldn't this be easier than twin primes? It just asks for gaps between semiprimes. arxiv.org/abs/math/0506067 may be useful. Products of two primes are tabulated at oeis.org/A001358 $\endgroup$
    – Gerry Myerson
    Commented Dec 23, 2017 at 19:13
  • $\begingroup$ So, is the answer to Question 1 yes, or still depending on some conjectures? $\endgroup$
    – Hailong Dao
    Commented Dec 23, 2017 at 19:46

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Goldston, Graham Pintz and Yildirim ( https://arxiv.org/abs/0803.2636 ) showed that among three linear forms $\ell_1, \ell_2, \ell_3$, which satisfy the obvious local conditions, there are two, say $\ell_1$ and $\ell_2$, such that there are infinitely many $m$, for which $\ell_1(m)$ and $\ell_2(m)$ are simultaneously prime. This implies that the set $\mathcal{N}$ of integers $n$, for which the equation $\det A= n$ is unsolvable with a matrix $A$ with prime entries doees not contain three integers $u, v, w$ with $u+v=w$. In particular the density of even integers for which the equation is unsolvable among all even integers is at most $\frac{1}{3}$. If you use several forms and the Zhang-Maynard-Tao method you can probably improve this result. However, proving that the equation is solvable for all even $n$ is probably difficult, and proving it for all $n$ is close to impossible (although not as impossible as 20 years ago).

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    $\begingroup$ I like "Not as impossible as 20 years ago". I never thought there could be different degrees of impossibility : -) $\endgroup$
    – Sylvain JULIEN
    Commented Dec 24, 2017 at 13:09

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