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On Certain Diophantine Equations Involving Lucas Numbers 论涉及卢卡斯数的某些二阶方程
arXiv - MATH - General Mathematics Pub Date : 2024-09-16 DOI: arxiv-2409.10152
Priyabrata Mandal
{"title":"On Certain Diophantine Equations Involving Lucas Numbers","authors":"Priyabrata Mandal","doi":"arxiv-2409.10152","DOIUrl":"https://doi.org/arxiv-2409.10152","url":null,"abstract":"This paper explores the intricate relationships between Lucas numbers and\u0000Diophantine equations, offering significant contributions to the field of\u0000number theory. We first establish that the equation regarding Lucas number $L_n\u0000= 3x^2$ has a unique solution in positive integers, specifically $(n, x) = (2,\u00001)$, by analyzing the congruence properties of Lucas numbers modulo $4$ and\u0000Jacobi symbols. We also prove that a Fibonacci number $F_n$ can be of the form\u0000$F_n=5x^2$ only when $(n,x)=(5,1)$. Expanding our investigation, we prove that\u0000the equation $L_n^2+L_{n+1}^2=x^2$ admits a unique solution $(n,x)=(2,5)$. In\u0000conclusion, we determine all non-negative integer solutions $(n, alpha, x)$ to\u0000the equation $L_n^alpha + L_{n+1}^alpha = x^2$, where $L_n$ represents the\u0000$n$-th term in the Lucas sequence.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"116 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Several formulae for summation over $SL(2,mathbb Z)$ 在 $SL(2,mathbb Z)$ 上求和的几个公式
arXiv - MATH - General Mathematics Pub Date : 2024-09-16 DOI: arxiv-2409.10592
Nikita Kalinin
{"title":"Several formulae for summation over $SL(2,mathbb Z)$","authors":"Nikita Kalinin","doi":"arxiv-2409.10592","DOIUrl":"https://doi.org/arxiv-2409.10592","url":null,"abstract":"New formulae for a summation over a positive part of $SL(2,mathbb Z)$ are\u0000presented. Such formulae can be written for any convex curve. We present\u0000several formulae where $pi$ is obtained.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252871","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Functional equation for Mellin transform of Fourier series associated with modular forms 与模数形式相关的傅里叶级数的梅林变换的函数方程
arXiv - MATH - General Mathematics Pub Date : 2024-09-10 DOI: arxiv-2409.06254
Omprakash Atale
{"title":"Functional equation for Mellin transform of Fourier series associated with modular forms","authors":"Omprakash Atale","doi":"arxiv-2409.06254","DOIUrl":"https://doi.org/arxiv-2409.06254","url":null,"abstract":"Let $X_1(s)$ and $X_2(s)$ denote the Mellin transforms of $chi_{1}(x)$ and\u0000$chi_{2}(x)$, respectively. Ramanujan investigated the functions $chi_1(x)$\u0000and $chi_2(x)$ that satisfy the functional equation begin{equation*}\u0000X_{1}(s)X_2(1-s) = lambda^2, end{equation*} where $lambda$ is a constant\u0000independent of $s$. Ramanujan concluded that elementary functions such as sine,\u0000cosine, and exponential functions, along with their reasonable combinations,\u0000are suitable candidates that satisfy this functional equation. Building upon\u0000this work, we explore the functions $chi_1(x)$ and $chi_2(x)$ whose Mellin\u0000transforms satisfy the more general functional equation begin{equation*}\u0000frac{X_1(s)}{X_2(k-s)} = sigma^2, end{equation*} where $k$ is an integer and\u0000$sigma$ is a constant independent of $s$. As a consequence, we show that the\u0000Mellin transform of the Fourier series associated to certain Dirichlet series\u0000and modular forms satisfy the same functional equation.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Finite Mellin Transform via Ramanujan's Master Theorem 通过拉曼努强主定理论有限梅林变换
arXiv - MATH - General Mathematics Pub Date : 2024-09-10 DOI: arxiv-2409.06304
Omprakash Atale
{"title":"On Finite Mellin Transform via Ramanujan's Master Theorem","authors":"Omprakash Atale","doi":"arxiv-2409.06304","DOIUrl":"https://doi.org/arxiv-2409.06304","url":null,"abstract":"This paper aims to show that by making use of Ramanujan's Master Theorem and\u0000the properties of the lower incomplete gamma function, it is possible to\u0000construct a finite Mellin transform for the function $f(x)$ that has infinite\u0000series expansions in positive integral powers of $x$. Some applications are\u0000discussed by evaluating certain definite integrals. The obtained solutions are\u0000also compared with results from Mathematica to test the validity of the\u0000calculations.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203930","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On infinite versions of the prisoner problem 关于囚徒问题的无限版本
arXiv - MATH - General Mathematics Pub Date : 2024-09-04 DOI: arxiv-2409.09064
Attila Losonczi
{"title":"On infinite versions of the prisoner problem","authors":"Attila Losonczi","doi":"arxiv-2409.09064","DOIUrl":"https://doi.org/arxiv-2409.09064","url":null,"abstract":"We investigate some versions of the famous 100 prisoner problem for the\u0000infinite case, where there are infinitely many prisoners and infinitely many\u0000boxes with labels. In this case, many questions can be asked about the\u0000admissible steps of the prisoners, the constraints they have to follow and also\u0000about the releasing conditions. We will present and analyze many cases. In the\u0000infinite case, the solutions and methods require mainly analysis rather than\u0000combinatorics.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Error estimation for numerical approximations of ODEs via composition techniques. Part I: One-step methods 通过组合技术估算 ODE 数值近似的误差。第一部分：一步法
arXiv - MATH - General Mathematics Pub Date : 2024-09-02 DOI: arxiv-2409.10548
Ahmad Deeb, Denys Dutykh
{"title":"Error estimation for numerical approximations of ODEs via composition techniques. Part I: One-step methods","authors":"Ahmad Deeb, Denys Dutykh","doi":"arxiv-2409.10548","DOIUrl":"https://doi.org/arxiv-2409.10548","url":null,"abstract":"In this study, we introduce a refined method for ascertaining error\u0000estimations in numerical simulations of dynamical systems via an innovative\u0000application of composition techniques. Our approach involves a dual application\u0000of a basic one-step numerical method of order p in this part, and for the class\u0000of Backward Difference Formulas schemes in the second part [Deeb A., Dutykh D.\u0000and AL Zohbi M. Error estimation for numerical approximations of ODEs via\u0000composition techniques. Part II: BDF methods, Submitted, 2024]. This dual\u0000application uses complex coefficients, resulting outputs in the complex plane.\u0000The methods innovation lies in the demonstration that the real parts of these\u0000outputs correspond to approximations of the solutions with an enhanced order of\u0000p + 1, while the imaginary parts serve as error estimations of the same order,\u0000a novel proof presented herein using Taylor expansion and perturbation\u0000technique. The linear stability of the resulted scheme is enhanced compared to\u0000the basic one. The performance of the composition in computing the\u0000approximation is also compared. Results show that the proposed technique\u0000provide higher accuracy with less computational time. This dual composition\u0000technique has been rigorously applied to a variety of dynamical problems,\u0000showcasing its efficacy in adapting the time step,particularly in situations\u0000where numerical schemes do not have theoretical error estimation. Consequently,\u0000the technique holds potential for advancing adaptive time-stepping strategies\u0000in numerical simulations, an area where accurate local error estimation is\u0000crucial yet often challenging to obtain.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An alternative proof of the Puiseux representation of the exponential integral 指数积分的 Puiseux 表示的另一种证明
arXiv - MATH - General Mathematics Pub Date : 2024-08-30 DOI: arxiv-2409.02949
Glenn Bruda
{"title":"An alternative proof of the Puiseux representation of the exponential integral","authors":"Glenn Bruda","doi":"arxiv-2409.02949","DOIUrl":"https://doi.org/arxiv-2409.02949","url":null,"abstract":"Working from definitions and an elementarily obtained integral formula for\u0000the Euler-Mascheroni constant, we give an alternative proof of the classical\u0000Puiseux representation of the exponential integral.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Metric dimensions of bicyclic graphs with potential applications in Supply Chain Logistics 双环图的度量维度及其在供应链物流中的潜在应用
arXiv - MATH - General Mathematics Pub Date : 2024-08-29 DOI: arxiv-2409.02947
Muwen Wang, Ghulam Haidar, Faisal Yousafzai, Murad Ul Islam Khan, Waseem Sikandar, Asad Ul Islam Khan
{"title":"Metric dimensions of bicyclic graphs with potential applications in Supply Chain Logistics","authors":"Muwen Wang, Ghulam Haidar, Faisal Yousafzai, Murad Ul Islam Khan, Waseem Sikandar, Asad Ul Islam Khan","doi":"arxiv-2409.02947","DOIUrl":"https://doi.org/arxiv-2409.02947","url":null,"abstract":"Metric dimensions and metric basis are graph invariants studied for their use\u0000in locating and indexing nodes in a graph. It was recently established that for\u0000bicyclic graph of type-III ($Theta $-graphs), the metric dimension is $3$\u0000only, when all paths have equal lengths, or when one of the outside path has a\u0000length $2$ more than the other two paths. In this article, we refute this claim\u0000and show that the case where the middle path is $2$ vertices more than the\u0000other two paths, also has metric dimension $3$. We also determine the metric\u0000dimension for other values of $p,q,r$ which were omitted in the recent research\u0000due to the constraint $p leq q leq r$. We also propose a graph-based\u0000technique to transform an agricultural supply chain logistics problem into a\u0000mathematical model, by using metric basis and metric dimensions. We provide a\u0000theoretical groundwork which can be used to model and solve these problems\u0000using machine learning algorithms.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Why and How the Definition of the Conformable Derivative in the Lower Terminal Should be Changed 为什么以及如何改变下端可变衍生物的定义
arXiv - MATH - General Mathematics Pub Date : 2024-08-26 DOI: arxiv-2409.02944
Hristo Kiskinov, Milena Petkova, Andrey Zahariev
{"title":"Why and How the Definition of the Conformable Derivative in the Lower Terminal Should be Changed","authors":"Hristo Kiskinov, Milena Petkova, Andrey Zahariev","doi":"arxiv-2409.02944","DOIUrl":"https://doi.org/arxiv-2409.02944","url":null,"abstract":"This paper discusses some unusual consequences raised by the definition of\u0000the conformable derivative in the lower terminal. A replacement for this\u0000definition is proposed and statements adjusted to the new definition are\u0000presented.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A note on friends of 20 关于 20 岁之友的说明
arXiv - MATH - General Mathematics Pub Date : 2024-08-25 DOI: arxiv-2409.04451
Tapas Chatterjee, Sagar Mandal, Sourav Mandal
{"title":"A note on friends of 20","authors":"Tapas Chatterjee, Sagar Mandal, Sourav Mandal","doi":"arxiv-2409.04451","DOIUrl":"https://doi.org/arxiv-2409.04451","url":null,"abstract":"Does $20$ have a friend? Or is it a solitary number? A folklore conjecture\u0000asserts that $20$ has no friends i.e. it is a solitary number. In this article,\u0000we prove that, a friend $N$ of $20$ is of the form $N=2cdot5^{2a}m^2$ and it\u0000has atleast six distinct prime divisors. Also we prove that $N$ must be atleast\u0000$2cdot 10^{12}$. Furthermore, we show that $Omega(N)geq 2omega(N)+6a-5$ and\u0000if $Omega(m)leq K$ then $N< 10cdot 6^{(2^{K-2a+3}-1)^2}$, where $Omega(n)$\u0000and $omega(n)$ denote the total number of prime divisors and the number of\u0000distinct prime divisors of the integer $n$ respectively. In addition, we deduce\u0000that, not all exponents of odd prime divisors of friend $N$ of $20$ are\u0000congruent to $-1$ modulo $f$, where $f$ is the order of $5$ in\u0000$(mathbb{Z}/pmathbb{Z})^times$ such that $3mid f$ and $p$ is a prime\u0000congruent to $1$ modulo $6$.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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