General classification and density of admissible integers

Establish general theorems that unify the modular-arithmetic observations for conjugate $(m,q)$-ary partitions, or, if complete classification of admissible integers is impossible, determine the density of integers that possess a conjugate $(m,q)$-ary partition.

Background

The paper derives congruence restrictions and analyzes several specific pairs $(m,q)$), especially $(4,3)$, but does not obtain a general characterization of all integers $n$ admitting a conjugate $(m,q)$-ary partition. The examples suggest that the relevant modular behavior depends subtly on the chosen bases.

The authors explicitly ask whether the observed phenomena can be unified into general theorems. They also propose density as a weaker alternative if exact classification for individual integers cannot be achieved.