  In this article, we study the classification of some natural numbers related
to the combinatorics of congruence subgroups of the modular group. More
precisely, we will focus here on the notion of minimal monomial solutions.
These are the solutions of a matrix equation (also appearing in the study of
Coxeter friezes), modulo an integer $N$, whose components are identical and
minimal for this property. Our objective here is to study the integers $N$ for
which the minimal monomial solutions satisfying some fixed conditions have an
irreducibility property. In particular, we will classify the monomially
irreducible integers which are the integers for which all the nonzero minimal
monomial solutions are irreducible.
