Rational approximation to algebraic numbers of small height: the Diophantine equation |axn - byn|= 1

MA Bennett - 2001‏ - degruyter.com
Following an approach originally due to Mahler and sharpened by Chudnovsky, we develop
an explicit version of the multi-dimensional``hypergeometric method''for rational and …

On integers with identical digits

Y Bugeaud, M Mignotte - Mathematika, 1999‏ - cambridge.org
On integers with identical digits Page 1 ON INTEGERS WITH IDENTICAL DIGITS YANN
BUGEAUD AND MAURICE MIGNOTTE §1. Introduction. A long-standing conjecture claims …

Exponential diophantine equations involving products of consecutive integers and related equations

TN Shorey - Number theory, 2000‏ - Springer
This paper contains an account of the results on the following topics: 1. Squares in products
from a block of consecutive integers 2. Equal products of consecutive integers 3. An …

[PDF][PDF] On the exponential Diophantine equation (12m2+ 1) x+(13m2− 1) y=(5m) z

N Terai, T Hibino - Int. J. Algebra, 2015‏ - m-hikari.com
On the Exponential Diophantine Equation (12m2 + 1)x + (13m2 − 1)y = (5m)z 1 Introduction
Page 1 International Journal of Algebra, Vol. 9, 2015, no. 6, 261 - 272 HIKARI Ltd, www.m-hikari.com …

[HTML][HTML] An upper bound for the number of solutions of ternary purely exponential diophantine equations

Y Hu, M Le - Journal of Number Theory, 2018‏ - Elsevier
Let a, b, c be fixed coprime positive integers with min⁡{a, b, c}> 1. In this paper, combining
the Gel'fond–Baker method with an elementary approach, we prove that if max⁡{a, b, c}> 5× …

On the exponential Diophantine equation (3pm^ 2-1)^ x+ (p (p-3) m^ 2+ 1)^ y=(pm)^ z (3 pm 2-1) x+(p (p-3) m 2+ 1) y=(pm) z

N Terai, T Hibino - Periodica Mathematica Hungarica, 2017‏ - Springer
Let m be a positive integer, and let p be a prime with p ≡ 1~(mod~ 4). p≡ 1 (mod 4). Then
we show that the exponential Diophantine equation (3pm^ 2-1)^ x+ (p (p-3) m^ 2+ 1) …

Perfect powers with few binary digits and related Diophantine problems, II

MA Bennett, Y Bugeaud, M Mignotte - Mathematical Proceedings of …, 2012‏ - cambridge.org
We prove that if q≥ 5 is an integer, then every qth power of an integer contains at least 5
nonzero digits in its binary expansion. This is a particular instance of one of a collection of …

A Note on the Exponential Diophantine Equation (rlm 2-1) x+(r (rl) m 2+ 1) y=(rm) z

S Fei, J Luo - Bulletin of the Brazilian Mathematical Society, New …, 2022‏ - Springer
Let r, l, m be positive integers such that 2∤ r, 3∣ l and r> l, and let e (r, l, m)= min {rlm 2-1, r
(rl) m 2+ 1}. In 2020, Kizildere, Le and Soydan proved if 3∤ rm and e (r, l, m)> 30, then the …

On the exponential Diophantine equation

T Miyazaki, N Terai - Bulletin of the Australian Mathematical Society, 2014‏ - cambridge.org
Let m, a, c be positive integers with a≡ 3, 5 (mod 8). We show that when 1+ c= a2, the
exponential Diophantine equation (m2+ 1) x+(cm2− 1) y=(am) z has only the positive integer …

The orbit problem in higher dimensions

V Chonev, J Ouaknine, J Worrell - Proceedings of the forty-fifth annual …, 2013‏ - dl.acm.org
We consider higher-dimensional versions of Kannan and Lipton's Orbit Problem---
determining whether a target vector space V may be reached from a starting point x under …