×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] M. Cipu, Y. Fujita and M. Mignotte, Two-parameter families of uniquely extendable Diophantine triples, Sci. China Math. 61 (2018), 421-438. · Zbl 1419.11054
[2] M. Cipu, Y. Fujita and T. Miyazaki, On the number of extensions of a Diophantine triple, Int. J. Number Theory 14 (2018), 899-917. · Zbl 1429.11065
[3] A. Dujella, Diophantine \(m\)-tuples, http://web.math.pmf.unizg.hr/ duje/dtuples.html. · Zbl 1046.11034
[4] A. Dujella and C. Fuchs, Complete solution of the polynomial version of a problem of Diophantus, J. Number Theory 106 (2004), 326-344. · Zbl 1047.11024
[5] A. Dujella and A. Jurasic, On the size of sets in a polynomial variant of a problem of Diophantus, Int. J. Number Theory 6 (2010), 1449-1471. · Zbl 1255.11011
[6] A. Dujella and F. Luca, On a problem of Diophantus with polynomials, Rocky Mountain J. Math. 37 (2007), 131-157. · Zbl 1145.11026
[7] A. Filipin and A. Jurasic, A polynomial variant of a problem of Diophantus and its consequences, arXiv:math/1705.09194. · Zbl 1455.11047
[8] Y. Fujita and T. Miyazaki, The regularity of Diophantine quadruples, Trans. Amer. Math. Soc. 370 (2018), 3803-3831. · Zbl 1417.11024
[9] B. He, A. Togbé and V. Ziegler, There is no Diophantine quintuple, Trans. Amer. Math. Soc., to appear. · Zbl 1430.11044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.