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

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