Coincidence between two binary recurrent sequences of polynomials arising from Diophantine triples.

*(English)*Zbl 1455.11025In 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)].

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)].

Reviewer: Mihai Cipu (Bucureşti)

##### Keywords:

binary recurrent sequences; Diophantine tuple; univariate polynomials with integer coefficients##### References:

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