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.