Proofs of the Conjectures on $SOME(n)$ and $DSOME(n)$ Functions Related to Integer Partitions

Abstract

Andrews and Dastidar (2026) introduced $SOME(n)$ and $DSOME(n)$ functions related to partitions of a positive integer $n$, where $SOME(n)$ is the sum of all the odd parts in the partitions of $n$ minus the sum of all the even parts and $DSOME(n)$ is the sum of all the odd parts in the partitions of $n$ into distinct parts minusthe sum of all the even parts in the same partitions. The purpose of this paper is to establish the conjecture$SOME(\lambda )\equiv 0(\mathrm{mod}{5}^{\alpha })$, $\alpha \ge 1$ and $\lambda \ge 0$ are integers such that $24\lambda \equiv 1(\mathrm{mod}{5}^{\alpha })$ due to Andrews and Dastidar, and the conjecture $DSOME(50n+21)\equiv 0(\mathrm{mod}8)$ due to Baruah and Gogoi (2026). In the process, we establish some new infinite families of congruences modulo 2, 4, and 8 for $DSOME(n)$.

25 pages