Some Observations on Modulo 5 Congruences for 2-Color Partitions

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).