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Year 2019, Volume: 48 Issue: 4, 1035 - 1045, 08.08.2019

Abstract

References

  • [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] B.U. Alfred, On square Lucas numbers, Fibonacci Quart. 2, 11–12, 1964.
  • [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] 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] 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] 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] J.H.E. Cohn, On square Fibonacci numbers, J. London Math. Soc. 39, 537–540, 1964.
  • [8] J.H.E. Cohn, Lucas and Fibonacci numbers and some Diophantine equations, Proc. Glasgow Math. Assoc. 7, 24–28, 1965.
  • [9] J.H.E. Cohn, Squares in some recurrent sequences, Pacific J. Math., 41, 631–646, 1972.
  • [10] J.H.E. Cohn, Perfect Pell powers, Glasgow Math. J. 38 (1), 19–20, 1996.
  • [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] R. Finkelstein, On Lucas numbers which are one more than a square, Fibonacci Quart. 13 (4), 340–342, 1975.
  • [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] 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] R. Keskin and Ü. Öğüt, Generalized Fibonacci Numbers of the form $wx^{2}+1$, Period. Math. Hungar. 73 (2), 165–178, 2016.
  • [16] T. Kovács, Combinatorial numbers in binary recurrences, Period. Math. Hungar. 58 (1), 83–98, 2009.
  • [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] W.L. McDaniel, Triangular numbers in the Pell sequence, Fibonacci Quart. 34 (2), 105–107, 1996.
  • [19] L. Ming, On triangular Fibonacci numbers, Fibonacci Quart., 27 (2), 98–108, 1989.
  • [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] L. Moser and L. Carlitz, Advanced problem H-2, Fibonacci Quart. 1, 46, 1963.
  • [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] A. Pethő, Full cubes in the Fibonacci sequence, Publ. Math. Debrecen, 30 (1-2), 117–127, 1983.
  • [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] P. Ribenboim, My Numbers, My Friends, Springer-Verlag, New York, Inc. 2000.
  • [26] P. Ribenboim and W.L. McDaniel, The square terms in Lucas sequences, J. Number Theory, 58, 104–123, 1996.
  • [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] N. Robbins, Fibonacci and Lucas numbers of the forms $w^{2}-1$, $w^{3}\pm1$, Fibonacci Quart. 19 (4), 369–373, 1981.
  • [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] A.P. Rollett, Advanced Problem No. 5080, Amer. Math. Monthly, February, 1963.
  • [31] R.J. Stroeker and N. Tzanakis, Computing all integer solutions of a genus 1 equation, Math. Comp. 72, 1917–1933, 2003.
  • [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] 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] Z. Şiar and R. Keskin, On square classes in generalized Fibonacci sequences, Acta Arith. 174 (3),277–295, 2016.
  • [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] N. Tzanakis, Elliptic Diophantine Equations (A concrete approach via the elliptic logarithm), Discrete Math. Appl. 195 pp, 2013.
  • [37] S. Tengely, Finding g-gonal numbers in recurrence sequences, Fibonacci Quart. 46/47 (3), 235–240, 2008/09.
  • [38] S. Tengely, On the Diophantine equation $L_{n}=\binom{x}{5}$, Publ. Math. Debrecen, 79 (3-4), 749–758, 2011.
  • [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.

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

Year 2019, Volume: 48 Issue: 4, 1035 - 1045, 08.08.2019

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 $P\equiv-1({mod}11).$

References

  • [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] B.U. Alfred, On square Lucas numbers, Fibonacci Quart. 2, 11–12, 1964.
  • [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] 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] 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] 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] J.H.E. Cohn, On square Fibonacci numbers, J. London Math. Soc. 39, 537–540, 1964.
  • [8] J.H.E. Cohn, Lucas and Fibonacci numbers and some Diophantine equations, Proc. Glasgow Math. Assoc. 7, 24–28, 1965.
  • [9] J.H.E. Cohn, Squares in some recurrent sequences, Pacific J. Math., 41, 631–646, 1972.
  • [10] J.H.E. Cohn, Perfect Pell powers, Glasgow Math. J. 38 (1), 19–20, 1996.
  • [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] R. Finkelstein, On Lucas numbers which are one more than a square, Fibonacci Quart. 13 (4), 340–342, 1975.
  • [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] 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] R. Keskin and Ü. Öğüt, Generalized Fibonacci Numbers of the form $wx^{2}+1$, Period. Math. Hungar. 73 (2), 165–178, 2016.
  • [16] T. Kovács, Combinatorial numbers in binary recurrences, Period. Math. Hungar. 58 (1), 83–98, 2009.
  • [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] W.L. McDaniel, Triangular numbers in the Pell sequence, Fibonacci Quart. 34 (2), 105–107, 1996.
  • [19] L. Ming, On triangular Fibonacci numbers, Fibonacci Quart., 27 (2), 98–108, 1989.
  • [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] L. Moser and L. Carlitz, Advanced problem H-2, Fibonacci Quart. 1, 46, 1963.
  • [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] A. Pethő, Full cubes in the Fibonacci sequence, Publ. Math. Debrecen, 30 (1-2), 117–127, 1983.
  • [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] P. Ribenboim, My Numbers, My Friends, Springer-Verlag, New York, Inc. 2000.
  • [26] P. Ribenboim and W.L. McDaniel, The square terms in Lucas sequences, J. Number Theory, 58, 104–123, 1996.
  • [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] N. Robbins, Fibonacci and Lucas numbers of the forms $w^{2}-1$, $w^{3}\pm1$, Fibonacci Quart. 19 (4), 369–373, 1981.
  • [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] A.P. Rollett, Advanced Problem No. 5080, Amer. Math. Monthly, February, 1963.
  • [31] R.J. Stroeker and N. Tzanakis, Computing all integer solutions of a genus 1 equation, Math. Comp. 72, 1917–1933, 2003.
  • [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] 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] Z. Şiar and R. Keskin, On square classes in generalized Fibonacci sequences, Acta Arith. 174 (3),277–295, 2016.
  • [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] N. Tzanakis, Elliptic Diophantine Equations (A concrete approach via the elliptic logarithm), Discrete Math. Appl. 195 pp, 2013.
  • [37] S. Tengely, Finding g-gonal numbers in recurrence sequences, Fibonacci Quart. 46/47 (3), 235–240, 2008/09.
  • [38] S. Tengely, On the Diophantine equation $L_{n}=\binom{x}{5}$, Publ. Math. Debrecen, 79 (3-4), 749–758, 2011.
  • [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.
There are 39 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Refik Keskin 0000-0003-2547-2082

Ümmügülsüm Öğüt This is me 0000-0002-5832-2382

Publication Date August 8, 2019
Published in Issue Year 2019 Volume: 48 Issue: 4

Cite

APA Keskin, R., & Öğüt, Ü. (2019). Generalized Lucas numbers of the form $11x^{2}\mp 1$. Hacettepe Journal of Mathematics and Statistics, 48(4), 1035-1045.
AMA Keskin R, Öğüt Ü. Generalized Lucas numbers of the form $11x^{2}\mp 1$. Hacettepe Journal of Mathematics and Statistics. August 2019;48(4):1035-1045.
Chicago Keskin, Refik, and Ümmügülsüm Öğüt. “Generalized Lucas Numbers of the Form $11x^{2}\mp 1$”. Hacettepe Journal of Mathematics and Statistics 48, no. 4 (August 2019): 1035-45.
EndNote Keskin R, Öğüt Ü (August 1, 2019) Generalized Lucas numbers of the form $11x^{2}\mp 1$. Hacettepe Journal of Mathematics and Statistics 48 4 1035–1045.
IEEE R. Keskin and Ü. Öğüt, “Generalized Lucas numbers of the form $11x^{2}\mp 1$”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 4, pp. 1035–1045, 2019.
ISNAD Keskin, Refik - Öğüt, Ümmügülsüm. “Generalized Lucas Numbers of the Form $11x^{2}\mp 1$”. Hacettepe Journal of Mathematics and Statistics 48/4 (August 2019), 1035-1045.
JAMA Keskin R, Öğüt Ü. Generalized Lucas numbers of the form $11x^{2}\mp 1$. Hacettepe Journal of Mathematics and Statistics. 2019;48:1035–1045.
MLA Keskin, Refik and Ümmügülsüm Öğüt. “Generalized Lucas Numbers of the Form $11x^{2}\mp 1$”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 4, 2019, pp. 1035-4.
Vancouver Keskin R, Öğüt Ü. Generalized Lucas numbers of the form $11x^{2}\mp 1$. Hacettepe Journal of Mathematics and Statistics. 2019;48(4):1035-4.