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Coincidence between two binary recurrent sequences of polynomials arising from Diophantine triples. (English) Zbl 1455.11025
In the ring $$R$$ of univariate polynomials with integer coefficients, a Diophantine set consists of finitely many polynomials $$a_i$$ ($$1\le i\le m$$), at least one of which has positive degree, such that $$a_1a_j+1$$ is a perfect square in $$R$$ for all $$1\le i < j\le m$$. A fruitful strategy for obtaining such sets is to enlarge a given Diophantine set by inserting an extra element without losing the defining property. The first difficult case is extension of Diophantine triples $$\{a,b,c\}$$, when the additional element $$d$$ is required to satisfy three quadratic equations with coefficients expressed in terms of $$a$$, $$b$$, and $$c$$. Thus, each suitable $$d$$ appears as a common term of two second order linear recurrent sequences $$d=v_m=w_n$$.
In the paper under review, the author finds all common terms of sequences having initial terms $$v_0$$, $$v_1$$, $$w_0$$, $$w_1$$ given by some explicit formulæ involving $$a$$, $$b$$, and $$c$$. The main theorem asserts that the equation $$v_m=w_n$$ has no solution with $$\min \{m,n\} > 2$$. This result is more general and its proof is simpler than that found in the paper by A. Dujella and C. Fuchs [J. Number Theory 106, No. 2, 326–344 (2004; Zbl 1047.11024)].
MSC:
 11B37 Recurrences 11D09 Quadratic and bilinear Diophantine equations
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References:
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