On a Diophantine equation of Erdős and Graham

doi:10.1016/j.jnt.2020.05.006

On a Diophantine equation of Erdős and Graham

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On a Diophantine equation of Erdős and Graham

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punktacja MEiN [2020]: 100

title:	On a Diophantine equation of Erdős and Graham
author:	Tengely Szabolcs, Ulas Maciej , Zygadło Jakub
journal title:	Journal of Number Theory
volume:	217
date of publication :	2020
pages:	445-459
ISSN:	0022-314X
eISSN:	1096-1658
DOI:	10.1016/j.jnt.2020.05.006
language:	English
journal language:	English
abstract in English:	We study solvability of the Diophantine equation in integers satisfying the conditions and for . The above Diophantine equation (of polynomial-exponential type) was mentioned in the monograph of Erdős and Graham, where several questions were stated. Some of these questions were already answered by Borwein and Loring. We extend their work and investigate other aspects of Erdős and Graham equation. First of all, we obtain the upper bound for the value given in terms of k only. This mean, that with fixed k our equation has only finitely many solutions in . Moreover, we construct an infinite set , such that for each , the considered equation has at least five solutions. As an application of our findings we enumerate all solutions of the equation for . Moreover, by applying greedy algorithm, we extend Borwein and Loring calculations and check that for each there is a value of k such that the considered equation has a solution in integers . Based on our numerical calculations we formulate some further questions and conjectures.
keywords in English:	polynomial exponential Diophantine equation, Erdős and Graham equation, Sum of fractions
affiliation:	Wydział Matematyki i Informatyki : Instytut Informatyki i Matematyki Komputerowej, Wydział Matematyki i Informatyki : Instytut Matematyki
type:	journal article
subtype:	academic paper