New congruences for 2-color partitions

TL;DR

This paper proves a conjecture on modulo 5 congruences for 2-color partition functions with specific restrictions and introduces new modulo 7 congruences for a special case, advancing understanding of partition congruences.

Contribution

It establishes a proof for a conjecture on modulo 5 congruences and presents new modulo 7 congruences for 2-color partitions, expanding the theory of partition congruences.

Findings

Proved a conjecture on modulo 5 congruences for p_k(n).

Discovered new modulo 7 congruences for p_4(n).

Enhanced understanding of partition congruences with color restrictions.

Abstract

Let $p_k(n)$ denote the number of $2$-color partitions of $n$ where one of the colors appears only in parts that are multiples of $k$. We will prove a conjecture of Ahmed, Baruah, and Dastidar on congruences modulo $5$ for $p_k(n)$. Moreover, we will present some new congruences modulo $7$ for $p_4(n)$.