Proof of a Conjecture on 6-colored Generalized Frobenius Partitions

Liuquan Wang

paper

Proof of a Conjecture on 6-colored Generalized Frobenius Partitions

arXiv:1507.03101

Abstract

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

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