In 2018, Luca and Patel conjectured that the largest perfect power representable as the sum of two Fibonacci numbers is 3864(2) = F-36 + F-12. In other words, they conjectured that the equation (*) y(a) = F-n + F(m)ha...
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In 2018, Luca and Patel conjectured that the largest perfect power representable as the sum of two Fibonacci numbers is 3864(2) = F-36 + F-12. In other words, they conjectured that the equation (*) y(a) = F-n + F(m)has no solutions with a >= 2 and y(a) > 3864(2). While this is still an open problem, there exist several partial results. For example, recently Kebli, Kihel, Larone and Luca proved an explicit upper bound for y(a), which depends on the size of y. In this paper, we find an explicit upper bound for y(a), which only depends on the Hamming weight of y with respect to the Zeckendorf representation. More specifically, we prove the following: If y = F-n1 + + F-nk and equation (*) is satisfied by y and some non-negative integers n, m and a >= 2, then y(a )<= exp (C(epsilon) k((3+epsilon)k2)) . Here, epsilon > 0 can be chosen arbitrarily and C(epsilon) is an effectively computable constant.
In this paper, we prove that there are finitely many multiplicative dependent vectors with coordinates from non-degenerate linear recurrence sequences {u(n)}(n >= 1-s) of a fixed order s >= 2. These sequences sa...
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In this paper, we prove that there are finitely many multiplicative dependent vectors with coordinates from non-degenerate linear recurrence sequences {u(n)}(n >= 1-s) of a fixed order s >= 2. These sequences satisfy the recurrence relation un = c(1)u(n-1 )+ c(2)u(n-2) + & ctdot;+ c(s)u(n-s) with initials u(1-s) = u(2-s) = & ctdot;= u(-1) = 0,u(0) = 1. Here, the coefficients of the recurrence relation are positive integers satisfying c(1) > 1 + c(2) + & ctdot;+ c(s) or c(1) >= c(2) >=& ctdot;>= c(s). In both these conditions, c(1) >= 4.
In [9], Gun, Murty and Rath studied non -vanishing and transcendental nature of special values of a varying class of L -functions and their derivatives. This led to a number of works by several authors in different se...
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In [9], Gun, Murty and Rath studied non -vanishing and transcendental nature of special values of a varying class of L -functions and their derivatives. This led to a number of works by several authors in different set-ups including studying higher derivatives. However, all these works were focused around the central point of the critical strip. In this article, we extend the study to arbitrary points in the critical strip. (c) 2024 Elsevier Inc. All rights reserved.
Suppose that ( U- n ) n >= 0 is a binary recurrence sequence and has a dominant root alpha with alpha > 1 and the discriminant D is square-free. In this paper, we study the Diophantine equation U- n + U- m = x (...
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Suppose that ( U- n ) n >= 0 is a binary recurrence sequence and has a dominant root alpha with alpha > 1 and the discriminant D is square-free. In this paper, we study the Diophantine equation U- n + U- m = x (q) in integers n >= m >= 0, x >= 2, and q >= 2. Firstly, we show that there are only finitely many of them for a fixed x using linear forms in logarithms. Secondly, we show that there are only finitely many solutions in ( n , m, x, q) with q, x >= 2 under the assumption of the abc-conjecture. To prove this, we use several classical results like Schmidt subspace theorem, a fundamental theorem on linear equations in S-units and Siegel's theorem concerning the finiteness of the number of solutions of a superelliptic equation.
In this study, we focus on finding the Pell and Pell-Lucas numbers which are concatenations of three repdigits. We show that these numbers are 169, 408, 985 and 198, 478, 1154, respectively. We use a lemma that provid...
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In this study, we focus on finding the Pell and Pell-Lucas numbers which are concatenations of three repdigits. We show that these numbers are 169, 408, 985 and 198, 478, 1154, respectively. We use a lemma that provides a large upper bound for the subscript n in the equations and Baker's theory of lower bounds for a nonzero linear form in logarithms of algebraic numbers. In addition, continued fraction expansions of some irrational numbers were calculated to show that some inequalities have no solution.
In this paper, we determine all Padovan numbers which can be written as concatenations of a Padovan number and a Perrin number. We find that all positive integer solutions to the Diophantine equation Pn=10dPm+Rk\docum...
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In this paper, we determine all Padovan numbers which can be written as concatenations of a Padovan number and a Perrin number. We find that all positive integer solutions to the Diophantine equation Pn=10dPm+Rk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{n}=10<^>{d}\cdot P_{m}+R_{k}$$\end{document} where m, n, k are nonnegative integers and d is the number of digits of Rk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{k}$$\end{document} are 12,37,151,351\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ 12,37,151,351\right\} $$\end{document}. Additionally, we find that all positive integer solutions to the Diophantine equation Pn=10dRm+Pk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{n}=10<^>{d}\cdot R_{m}+P_{k}$$\end{document} where m not equal 1,n,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ne 1,n,$$\end{document}k are nonnegative integers and d is the number of digits of Pk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathr
Let P(m )and E-m be them m-th Padovan and Perrin numbers, respectively. In this paper, we prove that for a fixed integer delta with delta >= 2 there exists finitely many Padovan and Perrin numbers that can be repre...
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Let P(m )and E-m be them m-th Padovan and Perrin numbers, respectively. In this paper, we prove that for a fixed integer delta with delta >= 2 there exists finitely many Padovan and Perrin numbers that can be represented as products of three repdigits in base ***, we explicitly find these numbers for 2 <= delta <= 10 as an application.
In this paper, among other things, we explicit a G delta -dense set of Liouville numbers, for which the triple power tower of any of its elements is a transcendental number. (c) 2023 Royal Dutch Mathematical Society (...
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In this paper, among other things, we explicit a G delta -dense set of Liouville numbers, for which the triple power tower of any of its elements is a transcendental number. (c) 2023 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.
It is an open question of Baker whether the numbers L(1, chi) for nontrivial Dirichlet characters x with period q are linearly independent over Q. The best known result is due to Baker, Birch and Wirsing which affirms...
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It is an open question of Baker whether the numbers L(1, chi) for nontrivial Dirichlet characters x with period q are linearly independent over Q. The best known result is due to Baker, Birch and Wirsing which affirms this when q is co-prime to phi(q). In this paper, we extend their result. to any arbitrary family of moduli. More precisely, for a positive integer q, let X-q denote the set of all L(1, chi) values as chi varies over nontrivial Dirichlet characters with period q. Then for any finite set of pairwise co-prime natural numbers q(i), 1 <= i <= l with (q(1) ... q(l), phi(q(1)) ... phi(q(l))) = I, we show that the set X-q1 U ... U X-ql is linearly independent over Q. In the process, we also extend a result of Okada about linear independence of the cotangent values over Q as well as a result of Murty-Murty about (Q) over bar linear independence of such L(1, chi) values. Finally, we prove Q linear independence of such L values of Erdosian functions with distinct prime periods p(i) for 1 <= i <= l with (P-1 ... P-l, (P-1 ... P-l)) = 1.
In this paper, we find all Padovan numbers which can be written as are difference of two repdigits. It is shown that all Padovan numbers which can be written as a difference of two repdigits are P-k is an element of {...
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In this paper, we find all Padovan numbers which can be written as are difference of two repdigits. It is shown that all Padovan numbers which can be written as a difference of two repdigits are P-k is an element of {2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 200, 3329}.
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