与整数分拆相关的$SOME(n)$和$DSOME(n)$函数的猜想证明
Proofs of the Conjectures on $SOME(n)$ and $DSOME(n)$ Functions Related to Integer Partitions
AI总结：
该文证明了与整数分拆相关的$SOME(n)$、$DSOME(n)$函数的两个已有同余猜想，同时得到了$DSOME(n)$模2、4、8的若干新同余无穷族。
AI中文摘要：
Andrews和Dastidar（2026）提出了与正整数$n$的分拆相关的$SOME(n)$和$DSOME(n)$函数，其中$SOME(n)$是$n$的所有分拆中的奇部之和减去偶部之和，$DSOME(n)$是$n$的所有互异分拆中的奇部之和减去同一分拆中的偶部之和。本文旨在证明Andrews和Dastidar提出的猜想$SOME(λ)≡0\bmod{5^α}$（其中$α≥1$、$λ≥0$为满足$24λ≡1\bmod{5^α}$的整数），以及Baruah和Gogoi（2026）提出的猜想$DSOME(50n+21)≡0\bmod{8}$。在证明过程中，我们建立了若干新的$DSOME(n)$模2、4、8的同余无穷族。
英文摘要：
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$SOME(λ)\equiv0\pmod{5^α}$, $α\ge 1$ and $λ\ge0$ are integers such that $24λ\equiv1\pmod{5^α}$ 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 $DSOME(n)$.
