Common values of linear recurrences related to Shank's simplest cubics

Pub Date : 2024-09-23 DOI:10.1016/j.jnt.2024.09.001
Attila Pethő , Szabolcs Tengely
{"title":"Common values of linear recurrences related to Shank's simplest cubics","authors":"Attila Pethő ,&nbsp;Szabolcs Tengely","doi":"10.1016/j.jnt.2024.09.001","DOIUrl":null,"url":null,"abstract":"<div><div>Let <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>∈</mo><mi>Z</mi></math></span> not all zeroes and let <span><math><mi>F</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>F</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> be the linear recursive sequence, which is defined by the initial terms <span><math><mi>F</mi><mo>(</mo><mi>u</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><mi>A</mi><mo>,</mo><mi>F</mi><mo>(</mo><mi>u</mi><mo>,</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>B</mi><mo>,</mo><mi>F</mi><mo>(</mo><mi>u</mi><mo>,</mo><mn>2</mn><mo>)</mo><mo>=</mo><mi>C</mi></math></span> and whose characteristic polynomial is Daniel Shanks simplest cubic <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><mo>(</mo><mi>u</mi><mo>−</mo><mn>1</mn><mo>)</mo><msup><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mo>(</mo><mi>u</mi><mo>+</mo><mn>2</mn><mo>)</mo><mi>X</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>u</mi><mo>∈</mo><mi>Z</mi></math></span>. We prove that there exists an effectively computable constant <em>c</em> depending only on <span><math><mi>L</mi><mo>=</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><mo>|</mo><mi>A</mi><mo>|</mo><mo>,</mo><mo>|</mo><mi>B</mi><mo>|</mo><mo>,</mo><mo>|</mo><mi>C</mi><mo>|</mo><mo>}</mo></math></span> such that if <span><math><mo>|</mo><mi>F</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>|</mo><mo>=</mo><mo>|</mo><mi>F</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>m</mi><mo>)</mo><mo>|</mo></math></span> holds for some integers <span><math><mi>u</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>m</mi></math></span> with <span><math><mi>n</mi><mo>≠</mo><mi>m</mi></math></span> then <span><math><mo>|</mo><mi>n</mi><mo>|</mo><mo>,</mo><mo>|</mo><mi>m</mi><mo>|</mo><mo>&lt;</mo><mi>c</mi></math></span>. For the choices <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>)</mo><mo>∈</mo><mo>{</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>}</mo></math></span> we solve the above equations completely. At the end we give an outlook to the equation <span><math><mi>F</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>u</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>F</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>v</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> for some fixed integers <span><math><mi>n</mi><mo>,</mo><mi>m</mi></math></span>.</div></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24001938","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

Let A,B,CZ not all zeroes and let F(u,n)=F(A,B,C,u,n) be the linear recursive sequence, which is defined by the initial terms F(u,0)=A,F(u,1)=B,F(u,2)=C and whose characteristic polynomial is Daniel Shanks simplest cubic Su(X)=X3(u1)X2(u+2)X1,uZ. We prove that there exists an effectively computable constant c depending only on L=max{|A|,|B|,|C|} such that if |F(A,B,C,u,n)|=|F(A,B,C,u,m)| holds for some integers u,n,m with nm then |n|,|m|<c. For the choices (A,B,C){(0,0,1),(1,1,1)} we solve the above equations completely. At the end we give an outlook to the equation F(0,0,1,u,n)=F(0,0,1,v,m) for some fixed integers n,m.
分享
查看原文
与尚克最简单立方体有关的线性递归的常见值
设 A,B,C∈Z 不全为零,并设 F(u,n)=F(A,B,C,u,n) 为线性递推序列,该序列由初始项 F(u,0)=A,F(u,1)=B,F(u,2)=C 定义,其特征多项式为丹尼尔-香克斯最简立方 Su(X)=X3-(u-1)X2-(u+2)X-1,u∈Z。我们证明存在一个有效的可计算常数 c,它只取决于 L=max{|A|,|B|,|C|},这样,如果|F(A,B,C,u,n)|=|F(A,B,C,u,m)|对某些整数 u,n,m 成立,且 n≠m,那么|n|,|m|<c。对于(A,B,C)∈{(0,0,1),(1,-1,1)}的选择,我们可以完全解出上述方程。最后,我们给出了对一些固定整数 n,m 的方程 F(0,0,1,u,n)=F(0,0,1,v,m) 的展望。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信