Proof of a Conjecture of Cui, Gu and Tang on 18-Colored Generalized Frobenius Partitions

RESUMO

Recently, the study of the number of $k$-colored generalized Frobenius partitions, denoted by $c\phi_k(n)$, has witnessed renewed interest. In this paper, we investigate congruence properties of $c\phi_{16}(n)$ and $c\phi_{18}(n)$. Our main result is a proof of the conjecture of Cui, Gu, and Tang \cite{CGT25} that, for all $n\ge0$, $c\phi_{18}(3n+2)\equiv0\pmod{2187}$. The proof uses a $(p,k)$-parametrization together with $q$-series identities and dissections. We also establish congruences for $c\phi_{16}(n)$ modulo $1024$ and $2048$, and for $c\phi_{18}(n)$ modulo $8$ and $81$.