<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>The aim of this paper is to consider the extensibility of the Diophantine triple {2, <italic>b, c</italic>}, where 2 <italic>&lt; b &lt; c</italic>, and to prove that such a set cannot be extended to an irregular Diophantine quadruple. We succeed in that for some families of <italic>c</italic>’s (depending on <italic>b</italic>). As corollary, for example, we prove that for <italic>b/</italic>2 − 1 prime, all Diophantine quadruples {2, <italic>b, c, d</italic>} with 2 <italic>&lt; b &lt; c &lt; d</italic> are regular.</p>
</abstract>

The aim of this paper is to consider the extensibility of the Diophantine triple {2, b, c}, where 2 < b < c, and to prove that such a set cannot be extended to an irregular Diophantine quadruple. We succeed in that for some families of c’s (depending on b). As corollary, for example, we prove that for b/2 − 1 prime, all Diophantine quadruples {2, b, c, d} with 2 < b < c < d are regular.

The extensibility of the Diophantine triple {2, <italic>b, c</italic>}

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

The extensibility of the Diophantine triple {2, }

The aim of this paper is to consider the extensibility of the Diophantine triple {2, b, c}, where 2 < b < c, and to prove that such a set cannot be extended to...

The extensibility of the Diophantine triple {2, b, c}

Published Online: Jul 08, 2021

Page range: 5 - 24

Received: Oct 14, 2020

Accepted: Nov 15, 2020

DOI: https://doi.org/10.2478/auom-2021-0016

Keywords
Diophantine tuples, system of Pellian equations, linear forms in logarithms

© 2021 Nikola Adžaga et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

The extensibility of the Diophantine triple {2, b, c}

Published Online: Jul 08, 2021

Page range: 5 - 24

Received: Oct 14, 2020

Accepted: Nov 15, 2020

DOI: https://doi.org/10.2478/auom-2021-0016

KeywordsDiophantine tuples, system of Pellian equations, linear forms in logarithms

© 2021 Nikola Adžaga et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Diophantine tuples, system of Pellian equations, linear forms in logarithms