การศึกษาปัญหาของผลแบ่งกั้นของจำนวนเต็มมอดุโล m โดยที่ฟังก์ชันตัวแทนเท่ากัน

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Published: Dec 27, 2023
Keywords:
partition representation function set of residue classes modulo m

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Supada Tokrasae
Master of Science Program in Teaching Mathematics, Division of Mathematics and Statistics, Faculty of Science and Technology, Nakhon Sawan Rajabhat University 60000, Thailand
ดร.นเรศ สวัสดิ์รักษา
Division of Mathematics and Statistics, Faculty of Science and Technology, Nakhon Sawan Rajabhat University 60000, Thailand
Somboon Niyom
Division of Mathematics and Statistics, Faculty of Science and Technology, Nakhon Sawan Rajabhat University 60000, Thailand
Sasisophit Buada
Division of Mathematics and Statistics, Faculty of Science and Technology, Nakhon Sawan Rajabhat University 60000, Thailand

Abstract

สำหรับทุกจำนวนเต็มบวก gif.latex?m กำหนดให้ gif.latex?\mathbb{Z}_{m} แทนเซตของชั้นส่วนตกค้างมอดุโล gif.latex?m ซึ่ง gif.latex?\mathbb{Z}_{m}&space;=&space;\left&space;\{&space;\bar{0},&space;\bar{1},&space;\bar{2},...,&space;\overline{m&space;-&space;1}&space;\right&space;\}  เมื่อ gif.latex?\bar{x}&space;=&space;\{&space;y&space;\in&space;\mathbb{Z}&space;:&space;x&space;\equiv&space;y&space;\(\textrm{mod}&space;m)&space;\} กำหนดนิยามอันดับ gif.latex?\prec บนเซต gif.latex?\mathbb{Z}_{m} ดังนี้ gif.latex?\bar{0}&space;\prec&space;\bar{1}&space;\prec&space;\bar{2}&space;\prec&space;\bar{3}...&space;\prec&space;\overline{m-1} และ gif.latex?\bar{a}&space;\preceq&space;\bar{b} ก็ต่อเมื่อ gif.latex?\bar{a}&space;=&space;\bar{b} หรือ gif.latex?\bar{a}&space;\prec&space;\bar{b} สำหรับเซตย่อย gif.latex?A&space;\subseteq&space;\mathbb{Z}_{m} และ gif.latex?\bar{n}\in&space;\mathbb{Z}_m กำหนดให้ 


gif.latex?R_{3}&space;(A,&space;\bar{n})&space;=&space;\left&space;|&space;\left&space;\{&space;(\bar{a},\bar{b})&space;\in&space;A&space;\times&space;A&space;:&space;a&space;+&space;b&space;\equiv&space;n(\mathrm{mod}&space;m),&space;\bar{a}&space;\preceq&space;\bar{b}&space;\right&space;\}&space;\right&space;|


ในงานวิจัยฉบับนี้ คณะผู้วิจัยศึกษาปัญหาของ Sárközy ใน gif.latex?\mathbb{Z}_{m} โดยที่เซตย่อย gif.latex?A และ gif.latex?B ของ gif.latex?\mathbb{Z}_{m} และ gif.latex?\left|&space;{\left(&space;{A&space;\cup&space;B}&space;\right)\backslash&space;\left(&space;{A&space;\cap&space;B}&space;\right)}&space;\right|&space;=&space;m&space;-&space;3  ที่ทำให้ gif.latex?{R_3}(A,\bar&space;n)&space;=&space;{R_3}(B,\bar&space;n) สำหรับทุก gif.latex?\bar{n}\in&space;\mathbb{Z}_m เมื่อ gif.latex?m เป็นจำนวนคี่

Article Details

Issue
Vol. 15 No. 22 (2023): JULY 2023 - DECEMBER 2023
Section
บทความวิจัย
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