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Mathematica Slovaca

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Volume 68, Issue 4

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Diophantine quadruples with values in k-generalized Fibonacci numbers

Carlos Alexis Gómez Ruiz / Florian Luca
  • School of Mathematics University of the Witwatersrand Private Bag X3 Wits 2050 South Africa
  • Max Planck Institute for Mathematics Vivatgasse 7, 53111 Bonn Germany
  • Department of Mathematics Faculty of Sciences University of Ostrava 30. Dubna 22, 701 03 Ostrava 1 Czech Republic
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Published Online: 2018-08-06 | DOI: https://doi.org/10.1515/ms-2017-0156

Abstract

We consider for integers k ≥ 2 the k–generalized Fibonacci sequences F(k) := (Fn(k))n2k, whose first k terms are 0, …, 0, 1 and each term afterwards is the sum of the preceding k terms. In this paper, we show that there does not exist a quadruple of positive integers a1 < a2 < a3 < a4 such that aiaj + 1 (ij) are all members of F(k).

MSC 2010: Primary 11B39; 11D61; Secondary 11B37

Keywords: generalized Fibonacci numbers; Diophantine quadruples

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About the article

C. A. G. thanks to the Universidad del Valle for support through Project 71079. F. L. was supported in part by grants CPRR160325161141 and an A-rated researcher award both from the NRF of South Africa and by grant no. 17-02804S of the Czech Granting Agency.


Received: 2016-01-24

Accepted: 2017-05-05

Published Online: 2018-08-06

Published in Print: 2018-08-28


Communicated by Federico Pellarin


Citation Information: Mathematica Slovaca, Volume 68, Issue 4, Pages 939–949, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0156.

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