The $ p $-Frobenius and $ p $-Sylvester numbers for Fibonacci and Lucas triplets

Author:

Komatsu Takao,Ying Haotian

Abstract

<abstract><p>In this paper we study a certain kind of generalized linear Diophantine problem of Frobenius. Let $ a_1, a_2, \dots, a_l $ be positive integers such that their greatest common divisor is one. For a nonnegative integer $ p $, denote the $ p $-Frobenius number by $ g_p (a_1, a_2, \dots, a_l) $, which is the largest integer that can be represented at most $ p $ ways by a linear combination with nonnegative integer coefficients of $ a_1, a_2, \dots, a_l $. When $ p = 0 $, the $ 0 $-Frobenius number is the classical Frobenius number. When $ l = 2 $, the $ p $-Frobenius number is explicitly given. However, when $ l = 3 $ and even larger, even in special cases, it is not easy to give the Frobenius number explicitly. It is even more difficult when $ p &gt; 0 $, and no specific example has been known. However, very recently, we have succeeded in giving explicit formulas for the case where the sequence is of triangular numbers <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup> or of repunits <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup> for the case where $ l = 3 $. In this paper, we show the explicit formula for the Fibonacci triple when $ p &gt; 0 $. In addition, we give an explicit formula for the $ p $-Sylvester number, that is, the total number of nonnegative integers that can be represented in at most $ p $ ways. Furthermore, explicit formulas are shown concerning the Lucas triple.</p></abstract>

Publisher

American Institute of Mathematical Sciences (AIMS)

Subject

Applied Mathematics,Computational Mathematics,General Agricultural and Biological Sciences,Modeling and Simulation,General Medicine

Cited by 11 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. p-Numerical Semigroups of Triples from the Three-Term Recurrence Relations;Axioms;2024-09-07

2. The p-Frobenius Number for the Triple of the Generalized Star Numbers;Symmetry;2024-08-22

3. Frobenius Numbers Associated with Diophantine Triples of x2-y2=zr;Symmetry;2024-07-05

4. On p-Frobenius of Affine Semigroups;Mediterranean Journal of Mathematics;2024-04-06

5. On the determination of p-Frobenius and related numbers using the p-Apéry set;Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas;2024-02-03

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3