Characterize Xi-matrices with composite determinant

Characterize the prime and composite matrices among non-dominated, adj-coprime \(2\times2\) nonnegative integer matrices whose determinant is composite, in order to obtain a structural classification beyond the available algorithmic test.

Background

The paper reduces the unresolved structural part of the primality problem to matrices in $\Xi$ with composite determinant. Earlier results characterize matrices outside $\Xi$, matrices with zero entries, matrices with determinant $1$, and several families determined by the arithmetic form of their determinant.

For a matrix in $\Xi$ with composite determinant, the paper provides a brute-force method and then an improved procedure when a candidate determinant of one factor is supplied. However, it does not provide a general structural characterization of which such matrices are prime.