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linear forms in logarithms and integral points on higher-dimensional varieties

ALGEBRA & NUMBER THEORY 2014年第3期8卷 647-687页

作者： Levin, Aaron Michigan State Univ Dept Math E Lansing MI 48824 USA

We apply inequalities from the theory of linear forms in logarithms to deduce effective results on S-integral points on certain higher-dimensional varieties when the cardinality of S is sufficiently small. These results may be viewed as a higher-dimensional version of an effective result of Bilu on integral points on curves. In particular, we prove a completely explicit result for integral points on certain affine subsets of the projective plane. As an application, we generalize an effective result of Vojta on the three-variable unit equation by giving an effective solution of the polynomial unit equation f(u, v) = w, where u, v, and w are S-units, vertical bar S vertical bar <= 3, and f is a polynomial satisfying certain conditions (which are generically satisfied). Finally, we compare our results to a higher-dimensional version of Runge's method, which has some characteristics in common with the results here.

关键词： integral points unit equation linear forms in logarithms Runge's method

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A KIT FOR linear forms IN THREE logarithms

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MATHEMATICS OF COMPUTATION 2024年第348期93卷 1903-1951页

作者： Mignotte, Maurice Voutier, Paul Laurent, Michel Univ Louis Pasteur UFR Math 7Rue Rene Descartes F-67084 Strasbourg France Aix Marseille Univ Inst Math Marseille 163 Ave LuminyCase 907 F-13288 Marseille 9 France

We provide a technique to obtain explicit bounds for problems that can be reduced to linear forms in three complex logarithms of algebraic numbers. This technique can produce bounds significantly better than general results on lower bounds for linear forms in logarithms. We give worked examples to demonstrate both the use of our technique and the improvements it provides. Publicly shared code is also available.

关键词： linear forms in logarithms diophantine equations

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A hypergeometric approach, via linear forms involving logarithms, to criteria for irrationality of Euler's constant

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MATHEMATICA SLOVACA 2009年第3期59卷 307-314页

作者： Sondow, Jonathan Zlobin, Sergey Moscow MV Lomonosov State Univ Fac Mech & Math Moscow 119899 Russia

Using an integral of a hypergeometric function, we give necessary and sufficient conditions for irrationality of Euler's constant gamma. The proof is by reduction to known irrationality criteria for gamma involving a Beukers-type double integral. We show that the hypergeometric and double integrals are equal by evaluating them. To do this, we introduce a construction of linear forms in 1, gamma, and logarithms from Nesterenko-type series of rational functions. In the Appendix, S. Zlobin gives a change-of-variables proof that the series and the double integral are equal.

关键词： Euler's constant irrationality hypergeometric linear forms in logarithms

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Lower bounds for linear forms in two p-adic logarithms

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JOURNAL OF NUMBER THEORY 2025年 266卷 295-349页

作者： Chim, Kwok Chi Graz Univ Technol Inst Anal & Number Theory Steyrergasse 30-2 A-8010 Graz Austria

We prove explicit lower bounds for linear forms in two p- adic logarithms. More specifically, we establish explicit lower bounds for the p-adic distance between two integral powers of algebraic numbers, that is, | Lambda | p = | alpha b 1 1 - alpha b 2 2 | p (and corresponding explicit upper bounds for v p ( Lambda )), where alpha 1 , alpha 2 are numbers that are algebraic over Q and b 1 , b 2 are positive rational integers. This work is a p-adic analogue of Gouillon's explicit lower bounds in the complex case. Our upper bound for v p ( Lambda ) has an explicit constant of reasonable size and the dependence of the bound on B (a quantity depending on b 1 and b 2 ) is log B , instead of (log B ) 2 as in the work of Bugeaud and Laurent in 1996. (c) 2024 The Author. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).

关键词： p-adic logarithmic form linear forms in logarithms Baker's method Approximation in non-Archimedean valuations

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Pell or Pell-Lucas numbers as concatenations of two repdigits in base b

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REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS 2025年第1期119卷 1-16页

作者： Adedji, Kouessi Norbert Filipin, Alan Rihane, Salah Eddine Togbe, Alain Univ Abomey Calavi Inst Math & Sci Phys Abomey Calavi Benin Univ Zagreb Fac Civil Engn Fra Andrije Kacica Miosica 26 Zagreb 10000 Croatia Natl Higher Sch Math Sidi Abdallah Algiers Algeria Purdue Univ Northwest Dept Math & Stat 2200 169th St Hammond IN 46323 USA

Let b be a positive integer such that b >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b\ge 2$$\end{document}. In this study, we prove that for a fixed b, there exists only a finite Pell and Pell-Lucas numbers as concatenations of two repdigits in base b. As a corollary, we show that the largest Pell or Pell-Lucas numbers which can be expressible as concatenations of two distinct repdigits in base b with 2 <= b <= 10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\le b\le 10$$\end{document} are P11=5741\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{11} = 5741$$\end{document} and Q5=82,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{5}=82,$$\end{document} respectively.

关键词： Pell numbers Pell-Lucas numbers b-repdigits linear forms in logarithms Diophantine equations Reduction method

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Effective approximation to complex algebraic numbers by quadratic numbers

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CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES 2025年第1期68卷 262-269页

作者： Bajpai, Prajeet Bugeaud, Yann Univ British Columbia Dept Math Vancouver BC V6T 1Z2 Canada Univ Strasbourg IRMA UMR 7501 7 rue Rene Descartes F-67084 Strasbourg France CNRS 7 rue Rene Descartes F-67084 Strasbourg France

We establish an effective improvement on the Liouville inequality for approximation to complex nonreal algebraic numbers by quadratic complex algebraic numbers.

关键词： Approximation to algebraic numbers linear forms in logarithms

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On Mulatu and generalized Lucas numbers which are products of four b-repdigits with a consequence

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INDIAN JOURNAL OF PURE & APPLIED MATHEMATICS 2025年 1-18页

作者： Adedji, Kouessi Norbert Univ Abomey Calavi UAC Inst Math et Sci Phys IMSP Abomey Calavi Benin

Let b be an integer such that b >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b\ge 2$$\end{document}. In this paper, we investigate on Mulatu and generalized Lucas numbers which are products of four repdigits in base b. Further on, we will fully determine these numbers for any b between 2 and 12 for Mulatu numbers and special cases of Lucas numbers namely Fibonacci, Pell and balancing as an application. As a corollary, we will discuss on intersection of Mulatu and generalized Lucas numbers. The proofs are based on Baker's theory on linear forms in logarithms of algebraic numbers and also the reduction method due to Bravo, G & oacute;mez and Luca.

关键词： Diophantine equations Generalized Lucas numbers Mulatu numbers b-Repdigits linear forms in logarithms Reduction method

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On the solutions of the Diophantine equation Fn ± Fm=3^s y^b

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INDIAN JOURNAL OF PURE & APPLIED MATHEMATICS 2025年 1-10页

作者： Siar, Zafer Erduran, Ibrahim Bingol Univ Dept Math Bingol Turkiye Berrin Topcuoglu Anatolian High Sch Gaziantep Turkiye

In this paper, we will give an upper bound on n satisfying the Diophantine equation F-n +/- F-m=3(s) y(b) in nonnegative integers s >= 0, y >= 2, b >= 2, n >= m > 0 and (y, 3) = 1. Then, we determine al... 详细信息

关键词： Fibonacci and Lucas numbers Exponential Diophantine equations linear forms in logarithms Baker's method

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On the resolution of the Diophantine equation U n + U m = x ^q

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RAMANUJAN JOURNAL 2025年第2期66卷 1-23页

作者： Bhoi, P. K. Rout, S. S. Panda, G. K. Natl Inst Technol Dept Math Rourkela 769008 India Natl Inst Technol Dept Math Kozhikode 673601 India

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.

关键词： Binary recurrence sequence Lucas sequences Diophantine equation linear forms in logarithms

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On Thabit and Williams numbers base b as sum or difference of Fibonacci and Mulatu numbers and vice versa

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AFRIKA MATEMATIKA 2025年第1期36卷 1-19页

作者： Adedji, Kouessi Norbert Bachabi, Mohamadou Togbe, Alain Univ Parakou UP Ecole Natl Stat Planificat et Demog ENSPD Parakou Benin Univ Abomey Calavi UAC Inst Math et Sci Phys IMSP Dangbo Benin Purdue Univ Northwest Dept Math & Stat 2200 169th St Hammond IN 46323 USA

Let Fnn >= 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( F_n\right) _{n \ge 0}$$\end{document} and Mnn >= 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( M_n\right) _{n \ge 0}$$\end{document} be the Fibonacci and Mulatu sequences. Let b be a positive integer such that b >= 2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b\ge 2.$$\end{document} In this paper, we prove that the following Diophantine equation Fn+& varepsilon;Mm=rho(b +/- 1)b & ell;+/- 1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F_n+\epsilon M_m=\rho \left( (b\pm 1)b<^>{\ell }\pm 1\right) , \end{aligned}$$\end{document}where (& varepsilon;,rho)is an element of{(1,1),(-1,1),(-1,-1)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\epsilon , \rho )\in \{(1, 1), (-1, 1), (-1, -1)\}$$\end{document} have only finitely many solutions in non-negative integers n, m, b and positive integer & ell;.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlen

关键词： Diophantine equations Thabit and Williams numbers Mulatu numbers Fibonacci numbers linear forms in logarithms Reduction method

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