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A problem on partitions of a residue class ring with the same representation | ||
| Journal of Algebra and Related Topics | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 14 آبان 1405 اصل مقاله (186.04 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2026.30956.1817 | ||
| نویسندگان | ||
| N. Sawatraksa* 1؛ N. Deein2 | ||
| 1Division of Mathematics and Statistics, Faculty of Science and Technology, Nakhon Sawan Rajabhat University, Thailand | ||
| 2Master of Science Program in Teaching Mathematics, Division of Mathematics and Statistics, Faculty of Science and Technology, Nakhon Sawan Rajabhat University, Thailand | ||
| چکیده | ||
| Let $ \mathbb{Z}_m $ denote the ring of residue classes modulo $m$. For $ A \subseteq \mathbb{Z}_m $, the representation $ R_A(\bar{n}) $ counts the number of solutions of equation $ a + {b} \equiv {n} \pmod{m} $ with $\bar{a}, \bar{b} \in A$. This paper investigates the conditions under which $ A, B \subseteq \mathbb{Z}_m $ satisfy $ R_A(\bar{n}) = R_B(\bar{n}) $ for all $ \bar{n} \in \mathbb{Z}_m $, under the constraint that $ |(A \cup B) \setminus (A \cap B)| = m - 4 $. We examine four structural cases based on the cardinality of the intersection $ A \cap B $, building upon earlier results for the case where $ |A \cap B| = 4$. We derive necessary structural conditions and establish nonexistence results for the remaining cases. The findings extend the classification of representation-equivalent subsets and contribute to ongoing studies of Sárközy’s problem in modular arithmetic. | ||
| کلیدواژه‌ها | ||
| Sárközy’s problem؛ Partition؛ Representation function؛ Set of residue classes modulo $ m $ | ||
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آمار تعداد مشاهده مقاله: 1 تعداد دریافت فایل اصل مقاله: 2 |
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