On a Diophantine equation of Erdős and Graham

2020
journal article
article
dc.abstract.enWe 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.pl
dc.affiliationWydział Matematyki i Informatyki : Instytut Informatyki i Matematyki Komputerowejpl
dc.affiliationWydział Matematyki i Informatyki : Instytut Matematykipl
dc.contributor.authorTengely, Szabolcspl
dc.contributor.authorUlas, Maciej - 147984 pl
dc.contributor.authorZygadło, Jakub - 114401 pl
dc.date.accessioned2020-09-24T14:32:31Z
dc.date.available2020-09-24T14:32:31Z
dc.date.issued2020pl
dc.date.openaccess0
dc.description.accesstimew momencie opublikowania
dc.description.physical445-459pl
dc.description.versionostateczna wersja wydawcy
dc.description.volume217pl
dc.identifier.doi10.1016/j.jnt.2020.05.006pl
dc.identifier.eissn1096-1658pl
dc.identifier.issn0022-314Xpl
dc.identifier.projectUMO-2019/34/E/ST1/00094pl
dc.identifier.projectROD UJ / OPpl
dc.identifier.urihttps://ruj.uj.edu.pl/xmlui/handle/item/246478
dc.languageengpl
dc.language.containerengpl
dc.rightsUdzielam licencji. Uznanie autorstwa 4.0 Międzynarodowa*
dc.rights.licenceCC-BY
dc.rights.urihttp://creativecommons.org/licenses/by/4.0/legalcode.pl*
dc.share.typeinne
dc.subject.enpolynomial exponential Diophantine equationpl
dc.subject.enErdős and Graham equationpl
dc.subject.enSum of fractionspl
dc.subtypeArticlepl
dc.titleOn a Diophantine equation of Erdős and Grahampl
dc.title.journalJournal of Number Theorypl
dc.typeJournalArticlepl
dspace.entity.typePublication
dc.abstract.enpl
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.
dc.affiliationpl
Wydział Matematyki i Informatyki : Instytut Informatyki i Matematyki Komputerowej
dc.affiliationpl
Wydział Matematyki i Informatyki : Instytut Matematyki
dc.contributor.authorpl
Tengely, Szabolcs
dc.contributor.authorpl
Ulas, Maciej - 147984
dc.contributor.authorpl
Zygadło, Jakub - 114401
dc.date.accessioned
2020-09-24T14:32:31Z
dc.date.available
2020-09-24T14:32:31Z
dc.date.issuedpl
2020
dc.date.openaccess
0
dc.description.accesstime
w momencie opublikowania
dc.description.physicalpl
445-459
dc.description.version
ostateczna wersja wydawcy
dc.description.volumepl
217
dc.identifier.doipl
10.1016/j.jnt.2020.05.006
dc.identifier.eissnpl
1096-1658
dc.identifier.issnpl
0022-314X
dc.identifier.projectpl
UMO-2019/34/E/ST1/00094
dc.identifier.projectpl
ROD UJ / OP
dc.identifier.uri
https://ruj.uj.edu.pl/xmlui/handle/item/246478
dc.languagepl
eng
dc.language.containerpl
eng
dc.rights*
Udzielam licencji. Uznanie autorstwa 4.0 Międzynarodowa
dc.rights.licence
CC-BY
dc.rights.uri*
http://creativecommons.org/licenses/by/4.0/legalcode.pl
dc.share.type
inne
dc.subject.enpl
polynomial exponential Diophantine equation
dc.subject.enpl
Erdős and Graham equation
dc.subject.enpl
Sum of fractions
dc.subtypepl
Article
dc.titlepl
On a Diophantine equation of Erdős and Graham
dc.title.journalpl
Journal of Number Theory
dc.typepl
JournalArticle
dspace.entity.type
Publication
Affiliations

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