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.