PreprintCC BY 4.0arXiv
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 cϕk(n), has witnessed renewed interest. In this paper, we investigate congruence properties of cϕ16(n) and cϕ18(n). Our main result is a proof of the conjecture of Cui, Gu, and Tang \cite{CGT25} that, for all n≥0, cϕ18(3n+2)≡0pmod2187. The proof uses a (p,k)-parametrization together with q-series identities and dissections. We also establish congruences for cϕ16(n) modulo $1024$ and $2048$, and for cϕ18(n) modulo $8$ and $81$.
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BibTeX
@misc{das2026proof,
title = {{Proof of a Conjecture of Cui, Gu and Tang on 18-Colored Generalized Frobenius Partitions}},
author = {Hirakjyoti Das and Hemjyoti Nath and Abhishek Sarma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05302},
eprint = {2609.05302},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05302},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Proof of a Conjecture of Cui, Gu and Tang on 18-Colored Generalized Frobenius Partitions
AU - Hirakjyoti Das
AU - Hemjyoti Nath
AU - Abhishek Sarma
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05302
UR - https://arxiv.org/abs/2609.05302
AB - 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≥0, cϕ_18(3n+2)≡0pmod2187. 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 $81$.
ER -