Proof of a Conjecture on 6-colored Generalized Frobenius Partitions

TL;DR

This paper proves a conjecture on divisibility properties of 6-colored generalized Frobenius partition functions, establishing new congruences and conjecturing further divisibility patterns.

Contribution

It confirms a specific divisibility conjecture and derives new congruences for the 6-colored Frobenius partition function, expanding understanding of its modular properties.

Findings

Proved that cφ₆(27n+16) ≡ 0 (mod 243).

Established a new congruence cφ₆(81n+61) ≡ 3 cφ₆(9n+7) (mod 243).

Identified multiple divisibility properties for cφ₆ at various moduli.

Abstract

Let $c\phi_{k}(n)$ be the $k$-colored generalized Frobenius partition function. By employing the generating function of $c\phi_{6}(3n+1)$ found by Hirschhorn, we prove that $c\phi_{6}(27n+16)\equiv 0$ (mod 243). This confirms a conjecture of E.X.W. Xia. We also find a congruence relation $c\phi_{6}(81n+61) \equiv 3 c\phi_{6}(9n+7)$ (mod 243). Moreover, we show that $c\phi_{6}(81n+61) \equiv 0$ (mod 81), $c\phi_{6}(243n+142) \equiv 0$ (mod 243) and $c\phi_{6}(729n+ 547) \equiv 0$ (mod 243). We further conjecture that for $n\ge 0$, $c\phi_{6}(243n+142) \equiv 0$ (mod 729).