This paper explores the intricate relationships between Lucas numbers and Diophantine equations, offering significant contributions to the field of number theory. We first establish that the equation regarding Lucas number $L_n = 3x^2$ has a unique solution in positive integers, specifically $(n, x) = (2, 1)$, by analyzing the congruence properties of Lucas numbers modulo $4$ and Jacobi symbols. We also prove that a Fibonacci number $F_n$ can be of the form $F_n=5x^2$ only when $(n,x)=(5,1)$. Expanding our investigation, we prove that the equation $L_n^2+L_{n+1}^2=x^2$ admits a unique solution $(n,x)=(2,5)$. In conclusion, we determine all non-negative integer solutions $(n, alpha, x)$ to the equation $L_n^alpha + L_{n+1}^alpha = x^2$, where $L_n$ represents the $n$-th term in the Lucas sequence.
{"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}
New formulae for a summation over a positive part of $SL(2,mathbb Z)$ are presented. Such formulae can be written for any convex curve. We present several formulae where $pi$ is obtained.
{"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}
Let $X_1(s)$ and $X_2(s)$ denote the Mellin transforms of $chi_{1}(x)$ and $chi_{2}(x)$, respectively. Ramanujan investigated the functions $chi_1(x)$ and $chi_2(x)$ that satisfy the functional equation begin{equation*} X_{1}(s)X_2(1-s) = lambda^2, end{equation*} where $lambda$ is a constant independent of $s$. Ramanujan concluded that elementary functions such as sine, cosine, and exponential functions, along with their reasonable combinations, are suitable candidates that satisfy this functional equation. Building upon this work, we explore the functions $chi_1(x)$ and $chi_2(x)$ whose Mellin transforms satisfy the more general functional equation begin{equation*} frac{X_1(s)}{X_2(k-s)} = sigma^2, end{equation*} where $k$ is an integer and $sigma$ is a constant independent of $s$. As a consequence, we show that the Mellin transform of the Fourier series associated to certain Dirichlet series and modular forms satisfy the same functional equation.
{"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}
This paper aims to show that by making use of Ramanujan's Master Theorem and the properties of the lower incomplete gamma function, it is possible to construct a finite Mellin transform for the function $f(x)$ that has infinite series expansions in positive integral powers of $x$. Some applications are discussed by evaluating certain definite integrals. The obtained solutions are also compared with results from Mathematica to test the validity of the calculations.
{"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}
We investigate some versions of the famous 100 prisoner problem for the infinite case, where there are infinitely many prisoners and infinitely many boxes with labels. In this case, many questions can be asked about the admissible steps of the prisoners, the constraints they have to follow and also about the releasing conditions. We will present and analyze many cases. In the infinite case, the solutions and methods require mainly analysis rather than combinatorics.
{"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}
In this study, we introduce a refined method for ascertaining error estimations in numerical simulations of dynamical systems via an innovative application of composition techniques. Our approach involves a dual application of a basic one-step numerical method of order p in this part, and for the class of Backward Difference Formulas schemes in the second part [Deeb A., Dutykh D. and AL Zohbi M. Error estimation for numerical approximations of ODEs via composition techniques. Part II: BDF methods, Submitted, 2024]. This dual application uses complex coefficients, resulting outputs in the complex plane. The methods innovation lies in the demonstration that the real parts of these outputs correspond to approximations of the solutions with an enhanced order of p + 1, while the imaginary parts serve as error estimations of the same order, a novel proof presented herein using Taylor expansion and perturbation technique. The linear stability of the resulted scheme is enhanced compared to the basic one. The performance of the composition in computing the approximation is also compared. Results show that the proposed technique provide higher accuracy with less computational time. This dual composition technique has been rigorously applied to a variety of dynamical problems, showcasing its efficacy in adapting the time step,particularly in situations where numerical schemes do not have theoretical error estimation. Consequently, the technique holds potential for advancing adaptive time-stepping strategies in numerical simulations, an area where accurate local error estimation is crucial yet often challenging to obtain.
在本研究中，我们通过创新性地应用组合技术，介绍了一种在动态系统数值模拟中确定误差估计的精炼方法。我们的方法在这一部分涉及 p 阶基本一步数值方法的双重应用，在第二部分则涉及一类后向差分公式方案[Deeb A., Dutykh D.and AL Zohbi M. Error estimation for numerical approximations of ODEs viacomposition techniques. Part II: BDF methods, Submitted, 2024]。第二部分：BDF 方法，已提交，2024 年]。该方法的创新之处在于证明了这些输出的实部对应于增强阶数为 p + 1 的解的近似值，而虚部则作为相同阶数的误差估计。与基本方案相比，结果方案的线性稳定性得到了增强。此外，还比较了计算近似值时的组成性能。结果表明，所提出的技术以更少的计算时间提供了更高的精度。这种二元组合技术已被严格应用于各种动力学问题，展示了它在调整时间步长方面的功效，尤其是在数值方案没有理论误差估计的情况下。因此，该技术具有在数值模拟中推进自适应时间步长策略的潜力，在这一领域，精确的局部误差估计至关重要，但往往难以获得。
{"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}
Working from definitions and an elementarily obtained integral formula for the Euler-Mascheroni constant, we give an alternative proof of the classical Puiseux representation of the exponential integral.
根据欧拉-马切洛尼常数的定义和元素积分公式，我们给出了指数积分的经典普伊索表示的另一种证明。
{"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}
Muwen Wang, Ghulam Haidar, Faisal Yousafzai, Murad Ul Islam Khan, Waseem Sikandar, Asad Ul Islam Khan
Metric dimensions and metric basis are graph invariants studied for their use in locating and indexing nodes in a graph. It was recently established that for bicyclic graph of type-III ($Theta $-graphs), the metric dimension is $3$ only, when all paths have equal lengths, or when one of the outside path has a length $2$ more than the other two paths. In this article, we refute this claim and show that the case where the middle path is $2$ vertices more than the other two paths, also has metric dimension $3$. We also determine the metric dimension for other values of $p,q,r$ which were omitted in the recent research due to the constraint $p leq q leq r$. We also propose a graph-based technique to transform an agricultural supply chain logistics problem into a mathematical model, by using metric basis and metric dimensions. We provide a theoretical groundwork which can be used to model and solve these problems using machine learning algorithms.
{"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}
This paper discusses some unusual consequences raised by the definition of the conformable derivative in the lower terminal. A replacement for this definition is proposed and statements adjusted to the new definition are presented.
{"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}
Does $20$ have a friend? Or is it a solitary number? A folklore conjecture asserts that $20$ has no friends i.e. it is a solitary number. In this article, we prove that, a friend $N$ of $20$ is of the form $N=2cdot5^{2a}m^2$ and it has atleast six distinct prime divisors. Also we prove that $N$ must be atleast $2cdot 10^{12}$. Furthermore, we show that $Omega(N)geq 2omega(N)+6a-5$ and if $Omega(m)leq K$ then $N< 10cdot 6^{(2^{K-2a+3}-1)^2}$, where $Omega(n)$ and $omega(n)$ denote the total number of prime divisors and the number of distinct prime divisors of the integer $n$ respectively. In addition, we deduce that, not all exponents of odd prime divisors of friend $N$ of $20$ are congruent to $-1$ modulo $f$, where $f$ is the order of $5$ in $(mathbb{Z}/pmathbb{Z})^times$ such that $3mid f$ and $p$ is a prime congruent to $1$ modulo $6$.
{"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}
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