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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)].
11B37 Recurrences
11D09 Quadratic and bilinear Diophantine equations
Full Text: DOI Euclid
[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
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