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December 2019 Coincidence Between Two Binary Recurrent Sequences of Polynomials Arising from Diophantine Triples
Takafumi MIYAZAKI
Tokyo J. Math. 42(2): 611-619 (December 2019). DOI: 10.3836/tjm/1502179292

Abstract

A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by 1 is a perfect square. A conjecture in this field asserts that any Diophantine triple can be uniquely extended to a Diophantine quadruple in some sense. This problem is reduced to study the coincidence between certain two binary recurrent sequences of integers. As an analogy of this, we consider a similar coincidence on the polynomial ring in one variable over the integers, and determine it completely. Our result is regarded as a generalization of a result in the paper “Complete solution of the polynomial version of a problem of Diophantus” by A. Dujella, C. Fuchs in Journal of Number Theory 106 (2004) on the polynomial variant of Diophantine tuples.

Citation

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Takafumi MIYAZAKI. "Coincidence Between Two Binary Recurrent Sequences of Polynomials Arising from Diophantine Triples." Tokyo J. Math. 42 (2) 611 - 619, December 2019. https://doi.org/10.3836/tjm/1502179292

Information

Published: December 2019
First available in Project Euclid: 8 March 2019

zbMATH: 07209636
MathSciNet: MR4106595
Digital Object Identifier: 10.3836/tjm/1502179292

Subjects:
Primary: 11B37
Secondary: 11D09

Rights: Copyright © 2019 Publication Committee for the Tokyo Journal of Mathematics

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Vol.42 • No. 2 • December 2019
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