Year: 2026
Type: preprint
Abstract: Recently, the study of the number of $k$-colored generalized Frobenius partitions, denoted by $cϕ_k(n)$, has witnessed renewed interest. In this paper, we investigate congruence properties of $cϕ_{16}(n)$ and $cϕ_{18}(n)$. Our main result is a proof of the conjecture of Cui, Gu, and Tang \cite{CGT25} that, for all $n\ge0$, $cϕ_{18}(3n+2)\equiv0\pmod{2187}$. The proof uses a $(p,k)$-parametrization together with $q$-series identities and dissections. We also establish congruences for $cϕ_{16}(n)$ modulo $1024$ and $2048$, and for $cϕ_{18}(n)$ modulo $8$ and $81$.
Source: arXiv (Cornell University)
Institutions: B. Borooah College, University of Florida
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7211884956