Solutions of Some Diophantine Equations in terms of Generalized Fibonacci and Lucas Numbers

Year 2019, Volume: 48 Issue: 2, 451 - 459, 01.04.2019

Bahar Demirtürk Bitim , Refik Keskin

Abstract

In this study, we present some identities involving generalized Fibonacci sequence $\left(U_{n}\right)$ and generalized Lucas sequence $\left(V_{n}\right)$. Then we give all solutions of the Diophantine equations $x^{2}-V_{n}xy+(-1)^{n}y^{2}=\pm (p^{2}+4)U_{n}^{2},$ $x^{2}-V_{n}xy+(-1)^{n}y^{2}=\pm U_{n}^{2},$ $x^{2}-(p^{2}+4)U_{n}xy-(p^{2}+4)(-1)^{n}y^{2}=\pm V_{n}^{2},$ $x^{2}-V_{n}xy\pm y^{2}=\pm 1,$ $x^{2}-(p^{2}+4)U_{n}xy-(p^{2}+4)(-1)^{n}y^{2}=1,$ $x^{2}-V_{n}xy+(-1)^{n}y^{2}=\pm (p^{2}+4)$, $x^{2}-V_{2n}xy+y^{2}=\pm(p^{2}+4)V_{n}^{2}$, $x^{2}-V_{2n}xy+y^{2}=(p^{2}+4)U_{n}^{2}$ and $x^{2}-V_{2n}xy+y^{2}=\pm V_{n}^{2}$ in terms of the sequences $\left( U_{n}\right) $ and $\left( V_{n}\right) $ with $p\geq 1$ and $p^{2}+4$ squarefree.

Keywords

generalized Fibonacci and Lucas sequences, Diophantine equations

References

• B. Demirtürk and R. Keskin, Integer Solutions of Some Diophantine Equations via Fibonacci and Lucas Numbers, J. Integer Seq. 12, 1-14, 2009.
• B. Demirtürk Bitim and R. Keskin, On Some Diophantine Equations, J. Inequal. Appl. 162, 1-12, 2013.
• P. Hilton and J. Pedersen, On generalized Fibonaccian and Lucasian numbers, Math. Gaz. 90 (518), 215-222, 2006.
• P. Hilton, J. Pedersen and L. Somer, On Lucasian numbers, Fibonacci Quart. 35, 43-47, 1997.
• D.E. Hinkel, An investigation of Lucas Sequences, Master thesis, Arizona University, 35 pages, 2007.
• A.F. Horadam, Basic properties of certain generalized sequences of numbers, Fi- bonacci Quart. 3 (3), 161-176, 1965.
• R. Keskin and B. Demirtürk, Solutions of Some Diophantine Equations Using Gen- eralized Fibonacci and Lucas Sequences, Ars Combin. 111, 161-179, 2013.
• E. Kılıç and N. Ömür, Conics characterizing the generalized Fibonacci and Lucas sequences with indices in arithmetic progressions, Ars Combin. 94, 459-464, 2010.
• P. Kiss, Diophantine representations of generalized Fibonacci numbers, Elem. Math. 34, 129-132, 1979.
• Y.V. Matiyasevich, Hilbert’s Tenth Problem, MIT Press, Cambridge, MA, 1993.
• W.L. McDaniel, Diophantine Representation of Lucas Sequences, Fibonacci Quart. 33, 58-63, 1995.
• R. Melham, Conics Which Characterize Certain Lucas Sequences, Fibonacci Quart. 35, 248-251, 1997.
• S. Rabinowitz, Algorithmic Manipulation of Fibonacci Identities, Applications of Fi- bonacci Numbers, 6 (edited by G. E. Bergum, et al.), Kluwer Academic Pub., Dor- drect, The Netherlands, 389-408, 1996.
• P. Ribenboim, Square classes of Fibonacci and Lucas numbers, Port. Math. 46 (2), 159-175, 1989.
• P. Ribenboim, The Little book of big primes, Springer-Verlag, New York, 1991.
• P. Ribenboim, An Algorithm to Determine the Points with Integral Coordinates in Certain Elliptic Curves, J. Number Theory 74, 19-38, 1999.
• P. Ribenboim, My numbers, My friends, Springer-Verlag Inc., New York, 2000.
• S. Zhiwei, Singlefold Diophantine representation of the sequence $U_{0}=0$, $U_{1}=1$ and $U_{n+2}=m U_{n+1}+U_{n}$, Pure and Applied Logic, Beijing Univ. Press, Beijing, 97-101, 1992.