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A KIT FOR linear forms IN THREE logarithms

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

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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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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 perfect powers that are difference of two Perrin numbers or two Padovan numbers

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PROCEEDINGS OF THE INDIAN NATIONAL SCIENCE ACADEMY 2024年第1期90卷 124-131页

作者： Duman, Merve Gueney Sakarya Univ Appl Sci Fac Technol Fundamental Sci Engn Sakarya Turkiye

Let (Pk)k >= 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_{k})_{k\ge 0}$$\end{document} be the sequence of Padovan numbers and (Rk)k >= 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(R_{k})_{k\ge 0}$$\end{document} be the sequence of Perrin numbers. In this paper, we solve the equations Rn-Rm=xa,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{n}-R_{m}=x<^>{a},$$\end{document}Pn-Pm=xa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{n}-P_{m}=x<^>{a}$$\end{document}, and Rn=xa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{n}=x<^>{a}$$\end{document} where n, m, a, x are nonnegative integers, 1 <= a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le a$$\end{document} and 2 <= x <= 10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemarg

关键词： Diophantine equations Continued fraction linear forms in logarithms Padovan number Perrin number

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Palindromic Concatenations of Two Distinct Repdigits in Narayana's Cows Sequence

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BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY 2024年第3期50卷 1-16页

作者： Ddamulira, Mahadi Emong, Paul Mirumbe, Geoffrey Ismail Makerere Univ Dept Math POB 7062 Kampala Uganda

Let (Nn)n >= 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(N_{n})_{n\ge 0}$$\end{document} be Narayana's cows sequence given by a recurrence relation Nn+3=Nn+2+Nn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ N_{n+3}=N_{n+2}+N_n $$\end{document} for all n >= 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ n\ge 0 $$\end{document}, with initial conditions N0=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ N_0=0 $$\end{document}, and N1=N2=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ N_1= N_2=1 $$\end{document}. In this paper, we find all members in Narayana's cows sequence that are palindromic concatenations of two distinct repdigits. Our proofs use techniques on Diophantine approximation which include the theory of linear forms in logarithms of algebraic numbers and Baker's reduction method. We show that 595 is the only member in Narayana's cow sequence that is a palindromic concatenation of two distinct repdigits in base 10.

关键词： Narayana's cows sequence Repdigit linear forms in logarithms Baker's method

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TRANSCENDENCE OF GENERALISED EULER-KRONECKER CONSTANTS

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BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY 2024年第3期109卷 464-470页

作者： Kandhil, Neelam Lunia, Rashi Max Planck Inst Math Vivatsgasse 7 D-53111 Bonn Germany Inst Math Sci CIT Campus Chennai 600113 India

We introduce some generalisations of the Euler-Kronecker constant of a number field and study their arithmetic nature.

关键词： linear forms in logarithms generalised Euler-Kronecker constants

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