Generalized Lucas numbers of the form $11x^{2}\mp 1$

Abstract

Let $P\geq3$ be an integer and $(V_{n})$ denote generalized Lucas sequence defined by $V_{0}=2,V_{1}=P,$ and $V_{n+1}=PV_{n}-V_{n-1}$ for $n\geq1.$ In this study, we solve the equation $V_{n}=11x^{2}\mp1.$ We show that the equation $V_{n}=11x^{2}+1$ has a solution only when $n=1$ and $P\equiv 1({mod}11)$. Moreover, we show that if the equation $V_{n}=11x^{2}-1$ has a solution, then $P\equiv2({mod}8)$ and $P\equiv-1({mod}11).$

Keywords

• generalized Lucas numbers,congruences,diophantine equation

References

1. [1] M.A. Alekseyev and S. Tengely, On integral points on biquadratic curves and nearmultiples of squares in Lucas sequences, J. Integer Seq. 17 Article ID: 14.6.6, 1–15, 2014.
2. [2] B.U. Alfred, On square Lucas numbers, Fibonacci Quart. 2, 11–12, 1964.
3. [3] A. Bremner and N. Tzanakis, Lucas sequences whose nth term is a square or an almost square, Acta Arith.126, 261–280, 2007.
4. [4] Y. Bugeaud, M. Mignotte, and S. Siksek, Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers, Ann. of Math. (2) 163 (3), 969–1018, 2006.
5. [5] Y. Bugeaud, M. Mignotte, F. Luca and S. Siksek, Fibonacci numbers at most one away from a perfect square, Elem. Math. 63, 65–75, 2008.
6. [6] W. Bosma, J. Cannon and C. Playoust, The MAGMA algebra system. I: The user language, J. Symbolic Comput. 24(3-4), 235–265, 1997.
7. [7] J.H.E. Cohn, On square Fibonacci numbers, J. London Math. Soc. 39, 537–540, 1964.
8. [8] J.H.E. Cohn, Lucas and Fibonacci numbers and some Diophantine equations, Proc. Glasgow Math. Assoc. 7, 24–28, 1965.

9. [9] J.H.E. Cohn, Squares in some recurrent sequences, Pacific J. Math., 41, 631–646, 1972.
10. [10] J.H.E. Cohn, Perfect Pell powers, Glasgow Math. J. 38 (1), 19–20, 1996.
11. [11] R. Finkelstein, On Fibonacci numbers which are one more than a square, Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, J. Reine Angew. Math. 262/263, 171–178, 1973.
12. [12] R. Finkelstein, On Lucas numbers which are one more than a square, Fibonacci Quart. 13 (4), 340–342, 1975.
13. [13] O. Karaatlı and R. Keskin, Generalized Lucas numbers of the form $5kx^{2}$ and $7kx^{2}$, Bull. Korean Math. Soc., 52 (5), 1467–1480, 2015.
14. [14] R. Keskin, Generalized Fibonacci and Lucas Numbers of the form $wx^{2}$ and $wx^{2}\mp1$, Bull. Korean Math. Soc. 51, 1041–1054, 2014.
15. [15] R. Keskin and Ü. Öğüt, Generalized Fibonacci Numbers of the form $wx^{2}+1$, Period. Math. Hungar. 73 (2), 165–178, 2016.
16. [16] T. Kovács, Combinatorial numbers in binary recurrences, Period. Math. Hungar. 58 (1), 83–98, 2009.
17. [17] F. Luca, Fibonacci numbers of the form $k^{2}+k+2$, in: Applications of Fibonacci numbers, 8 (Rochester, NY, 1998), 241–249, Kluwer Acad. Publ. Dordrecht, 1999.
18. [18] W.L. McDaniel, Triangular numbers in the Pell sequence, Fibonacci Quart. 34 (2), 105–107, 1996.
19. [19] L. Ming, On triangular Fibonacci numbers, Fibonacci Quart., 27 (2), 98–108, 1989.
20. [20] L. Ming, On triangular Lucas numbers, in: Applications of Fibonacci numbers, 4, (Winston-Salem, NC, 1990), 231–240, Kluwer Acad. Publ. Dordrecht, 1991.
21. [21] L. Moser and L. Carlitz, Advanced problem H-2, Fibonacci Quart. 1, 46, 1963.
22. [22] Ü. Öğüt and R. Keskin, Generalized Fibonacci and Lucas Numbers of the form $11x^{2}+1$, Honam Math. J. 40 (1), 139–153, 2018.
23. [23] A. Pethő, Full cubes in the Fibonacci sequence, Publ. Math. Debrecen, 30 (1-2), 117–127, 1983.
24. [24] A. Pethő, The Pell sequence contains only trivial perfect powers, in: Sets, graphs and numbers (Budapest, 1991), 561–568, Colloq. Math. Soc. János Bolyai, 60, North- Holland, Amsterdam, 1992.
25. [25] P. Ribenboim, My Numbers, My Friends, Springer-Verlag, New York, Inc. 2000.
26. [26] P. Ribenboim and W.L. McDaniel, The square terms in Lucas sequences, J. Number Theory, 58, 104–123, 1996.
27. [27] P. Ribenboim and W.L. McDaniel, On Lucas sequence terms of the form $kx^{2}$, Number Theory: proceedings of the Turku symposium on Number Theory in memory of Kustaa Inkeri (Turku, 1999), Walter de Gruyter, Berlin, 293–303, 2001.
28. [28] N. Robbins, Fibonacci and Lucas numbers of the forms $w^{2}-1$, $w^{3}\pm1$, Fibonacci Quart. 19 (4), 369–373, 1981.
29. [29] N. Robbins, Fibonacci numbers of the forms $px^{2}\pm1$, $px^{3}\pm1$, where p is prime, in: Applications of Fibonacci numbers (San Jose, CA, 1986), 77–88, Kluwer Acad. Publ. Dordrecht, 1988.
30. [30] A.P. Rollett, Advanced Problem No. 5080, Amer. Math. Monthly, February, 1963.
31. [31] R.J. Stroeker and N. Tzanakis, Computing all integer solutions of a genus 1 equation, Math. Comp. 72, 1917–1933, 2003.
32. [32] Z. Şiar and R. Keskin, Some new identities concerning generalized Fibonacci and Lucas numbers, Hacet. J. Math. Stat. 42 (3), 211–222, 2013.
33. [33] Z. Şiar and R. Keskin, The Square terms in generalized Lucas sequence with parameters P and Q, Math. Scand. 118 (1), 13–26, 2016.
34. [34] Z. Şiar and R. Keskin, On square classes in generalized Fibonacci sequences, Acta Arith. 174 (3),277–295, 2016.
35. [35] N. Tzanakis, Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms. The case of quartic equations, Acta Arith. 75, 165–190, 1996.
36. [36] N. Tzanakis, Elliptic Diophantine Equations (A concrete approach via the elliptic logarithm), Discrete Math. Appl. 195 pp, 2013.
37. [37] S. Tengely, Finding g-gonal numbers in recurrence sequences, Fibonacci Quart. 46/47 (3), 235–240, 2008/09.
38. [38] S. Tengely, On the Diophantine equation $L_{n}=\binom{x}{5}$, Publ. Math. Debrecen, 79 (3-4), 749–758, 2011.
39. [39] O. Wyler, In the Fibonacci series $F_{1}=1,F_{2}=1,F_{n+1}=F_{n}+F_{n-1}$ the first, second and twelfth terms are squares, Amer. Math. Monthly, 71, 221–222, 1964.