Congruences modulo powers of $3$ for $6$-colored generalized Frobenius partitions
Dandan Chen, Siyu Yin
TL;DR
The paper proves congruences modulo powers of $3$ for $c\phi_6(n)$ at arguments of the form $3^{\alpha}n+\lambda_{\alpha}$, solving a revised conjecture of Gu, Wang, and Xia. It combines a localization method with modular equations on $\Gamma_0(12)$ and an eta-quotient framework, employing $U$-operators and detailed recurrences to control the $3$-adic valuation of generating-function coefficients. The authors establish that $c\phi_6(3^{\alpha}n+\lambda_{\alpha})\equiv0\pmod{3^{\lfloor\alpha/2\rfloor+2}}$ for all $\alpha\ge1$, with explicit $\lambda$-parameters separating odd and even $\alpha$. This advances the understanding of partition congruences in the $6$-colored generalized Frobenius partition family and demonstrates a robust technique for higher-power modulus results in the theory of modular forms and partitions.
Abstract
In 1984, Andrews introduced the family of partition functions $cφ_k(n)$, the number of generalized Frobenius partitions of $n$ with $k$ colors. In 2016, Gu, Wang, and Xia proved some congruences about $cφ_6(n)$ and gave a conjecture on congruences modulo powers of 3 for $cφ_6(n)$. We solve the revised conjecture proposed by Gu, Wang, and Xia using a method similar to that of Banerjee and Smoot.