On the $x$-coordinates of Pell equations which are Fibonacci numbers

  • Florian Luca
  • Alain Togbé

Abstract

For an integer $d>2$ which is not a square, we show that there is at most one value of the positive integer $x$ participating in the Pell equation $x^2-dy^2=\pm 1$ which is a Fibonacci number.

References

Bugeaud, Y., Mignotte, M., and Siksek, S., Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers, Ann. of Math. (2) 163 (2006), no. 3, 969–1018. https://doi.org/10.4007/annals.2006.163.969

>

Carmichael, R. D., On the numerical factors of the arithmetic forms $alpha ^npm beta ^n$, Ann. of Math. (2) 15 (1913/14), no. 1-4, 30–70. https://doi.org/10.2307/1967797

>

Cohn, J. H. E., On square Fibonacci numbers, J. London Math. Soc. 39 (1964), 537–540. https://doi.org/10.1112/jlms/s1-39.1.537

>

Cohn, J. H. E., The Diophantine equation $x^4-Dy^2=1$, Quart. J. Math. Oxford Ser. (2) 26 (1975), no. 1, 279–281. https://doi.org/10.1093/qmath/26.1.279

>

Cohn, J. H. E., The Diophantine equation $x^4-Dy^2=1$. II, Acta Arith. 78 (1997), no. 4, 401–403. https://doi.org/10.4064/aa-78-4-401-403

>

Dujella, A. and Pethő, A., A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2) 49 (1998), no. 3, 291–306. https://doi.org/10.1093/qjmath/49.195.291

>

Kalman, D. and Mena, R., The Fibonacci numbers—exposed, Math. Mag. 76 (2003), no. 3, 167–181. https://doi.org/10.2307/3219318

>

Ljunggren, W., Über die unbestimmte Gleichung $Ax^2 - By^4 = C$, Arch. Math. Naturvid. 41 (1938), no. 10, 18.

Ljunggren, W., Über die Gleichung $x^4-Dy^2=1$, Arch. Math. Naturvid. 45 (1942), no. 5, 61–70.

Ljunggren, W., On the Diophantine equation $x^2+4=Ay^4$, Norske Vid. Selsk. Forh., Trondheim 24 (1951), 82–84.

Ljunggren, W., Collected papers of Wilhelm Ljunggren. Vol. 1, 2, Queen's Papers in Pure and Applied Mathematics, vol. 115, Queen's University, Kingston, ON, 2003.

Matveev, E. M., An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), no. 6, 125–180. https://doi.org/10.1070/IM2000v064n06ABEH000314

>

McDaniel, W. L., The g.c.d. in Lucas sequences and Lehmer number sequences, Fibonacci Quart. 29 (1991), no. 1, 24–29.

Ming, L., On triangular Fibonacci numbers, Fibonacci Quart. 27 (1989), no. 2, 98–108.

Posamentier, A. S. and Lehmann, I., The (fabulous) Fibonacci numbers, Prometheus Books, Amherst, NY, 2007.

Rollett, A. P. and Wyler, O., Advanced Problems and Solutions: Solutions: 5080, Amer. Math. Monthly 71 (1964), no. 2, 220–222.

Published
2018-02-20
How to Cite
Luca, F., & Togbé, A. (2018). On the $x$-coordinates of Pell equations which are Fibonacci numbers. MATHEMATICA SCANDINAVICA, 122(1), 18-30. https://doi.org/10.7146/math.scand.a-97271
Section
Articles