## Tokyo Journal of Mathematics

### Coincidence Between Two Binary Recurrent Sequences of Polynomials Arising from Diophantine Triples

Takafumi MIYAZAKI

#### 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.

#### Article information

Source
Tokyo J. Math., Volume 42, Number 2 (2019), 611-619.

Dates
First available in Project Euclid: 8 March 2019

https://projecteuclid.org/euclid.tjm/1552013764

#### Citation

MIYAZAKI, Takafumi. Coincidence Between Two Binary Recurrent Sequences of Polynomials Arising from Diophantine Triples. Tokyo J. Math. 42 (2019), no. 2, 611--619. https://projecteuclid.org/euclid.tjm/1552013764

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