Rad HAZU, Matematičke znanosti, Vol. 26 (2022), 21-43.

THE PROBLEM OF THE EXTENSION OF D(4)-TRIPLE {1, b, c}

Kouèssi Norbert Adédji, Alan Filipin and Alain Togbé

Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića-Miošića 26, 10000 Zagreb, Croatia
e-mail: alan.filipin@grad.unizg.hr

Department of Mathematics, Statistics and Computer Science, Purdue University Northwest, 1401 S, U.S. 421, Westville IN 46391 USA
e-mail: atogbe@pnw.edu

Abstract.   In this paper, we study the extensibility of the D(4)-triple {1, b, c}, where 1 < b < c, by proving that such a set cannot be extended to an irregular D(4)-quadruple only for some values of c. For this study, we will use the classical methods based on the resolution of the binary recurrence sequences with new approaches in order to confirm a conjecture of uniqueness of such an extension.

2020 Mathematics Subject Classification.   11D09, 11B37, 11J68, 11J86, 11Y65.

Key words and phrases.   Diophantine m-tuples, system of Pellian equations, reduction method, linear form in logarithms.

DOI: https://doi.org/10.21857/ygjwrcpgoy/

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