Some Observations on Modulo 5 Congruences for 2-Color Partitions

TL;DR

This paper characterizes modulo 5 congruences for 2-color partition functions, providing complete results, simple proofs for some cases, counter-examples for others, and an alternative proof for a specific case, enhancing understanding of these partition congruences.

Contribution

It offers a complete characterization of modulo 5 congruences for 2-color partition functions, including new proofs and counter-examples for various cases.

Findings

Complete characterization of $p_k(25n + 24 - k)$ modulo 5

Simple proofs for specific values of $k$

Counter-examples for remaining cases

Abstract

The 2-color partitions may be considered as an extension of regular partitions of a natural number $n$, with $p_{k}(n)$ defined as the number of 2-colored partitions of $n$ where one of the 2 colors appears only in parts that are multiples of $k$. In this paper, we record the complete characterization of the modulo 5 congruence relation $p_{k}(25n + 24 - k) \equiv 0 \pmod{5}$ for $k \in \{1, 2, \ldots, 24\}$, in connection with the 2-color partition function $p_k(n)$, providing references to existing results for $k \in \{1, 2, 3, 4, 7, 8, 17\}$, simple proofs for $k \in \{5, 10, 15, 20\}$ for the sake of completeness, and counter-examples in all the remaining cases. We also propose an alternative proof in the case of $k = 4$, without using the Rogers-Ramanujan ratio, thereby making the proof considerably simpler compared to the proof by Ahmed, Baruah and Ghosh Dastidar (JNT 2015).