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arXiv · 2608.08524

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

Gaurab Bardhan·Nipen Saikia
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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$SOME(\lambda)\equiv0\pmod{5^\alpha}$, $\alpha\ge 1$ and $\lambda\ge0$ are integers such that $24\lambda\equiv1\pmod{5^\alpha}$ due to Andrews and Dastidar, and the conjecture $DSOME(50n+21)\equiv0\pmod{8}$ due to Baruah and Gogoi (2026). In the process, we establish some new infinite families of congruences modulo 2, 4, and 8 for $DSOME(n)$.

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Gaurab Bardhan, Nipen Saikia. 2026-08-09. Proofs of the Conjectures on $SOME(n)$ and $DSOME(n)$ Functions Related to Integer Partitions. https://arxiv.org/abs/2608.08524

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