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 $c{\varphi }_k(n)$, has witnessed renewed interest. In this paper, we investigate congruence properties of $c{\varphi }_{16}(n)$ and $c{\varphi }_{18}(n)$. Our main result is a proof of the conjecture of Cui, Gu, and Tang \cite{CGT25} that, for all $n\ge 0$, $c{\varphi }_{18}(3n+2)\equiv 0(\mathrm{mod}2187)$. The proof uses a $(p,k)$-parametrization together with $q$-series identities and dissections. We also establish congruences for $c{\varphi }_{16}(n)$ modulo $1024$ and $2048$, and for $c{\varphi }_{18}(n)$ modulo $8$ and $81$.

22 pages