Glasnik Matematicki, Vol. 55, No. 1 (2020), 1-12.

ON Y-COORDINATES OF PELL EQUATIONS WHICH ARE BASE 2 REP-DIGITS

Bernadette Faye-Fall and Florian Luca

UFR SAT, Université Gaston Berger , Saint-Louis 32002, Sénégal
e-mail: bernadette.fayee@gmail.com

School of Mathematics, University of the Witwatersrand, Private Bag X3, Wits 2050, Johannesburg, South Africa,
and
Research Group in Algebraic Structures and Applications, King Abdulaziz University, Jeddah, Saudi Arabia,
and
Centro de Ciencias Matemáticas, UNAM, Morelia, Mexico
e-mail: florian.luca@wits.ac.za

Abstract.   In this paper, we show that if (Xk,Yk) is the kth solution of the Pell equation X2-dY2=1 for some non-square integer d>1, then the equation Yk=2n-1 has at most two positive integer solutions (k,n).

2010 Mathematics Subject Classification.   11B39, 11D61

Key words and phrases.   Pell equations, exponential Diophantine equations, applications of linear forms in logarithms

Full text (PDF) (access from subscribing institutions only)

https://doi.org/10.3336/gm.55.1.01

References:

  1. A. Baker and H. Davenport, The equations 3x2 - 2= y2 and 8x2 - 7= z2, Q. J. Math. 20 (1969), 129-137.
    MathSciNet     CrossRef

  2. M. A. Bennett, On the number of solutions of simultaneous Pell equations, J. Reine Angew. Math. 498 (1998), 173-199.
    MathSciNet     CrossRef

  3. J. H. E. Cohn, The Diophantine equation x4-Dy2=1, II, Acta Arith. 78 (1997), 401-403.
    MathSciNet     CrossRef

  4. A. Dossavi-Yovo, F. Luca and A. Togbé, On the x-coordinates of Pell equations which are rep-digits, Publ. Math. Debrecen 88 (2016), 381-399.
    MathSciNet     CrossRef

  5. H. S. Erazo, C. A. Gómez and F. Luca, On Pillai's problem with X-coordinates of Pell equations and powers of 2, J. Number Theory 203 (2019), 294-309.
    MathSciNet     CrossRef

  6. B. Faye and F. Luca, On the X-coordinates of Pell equations which are rep-digits, Fibonacci Quart. 56 (2018), 52-62.
    MathSciNet    

  7. S. Laisharam, F. Luca and M. Sias, On the Diophantine equation n!2± 1=dv2, preprint, 2018.

  8. W. Ljunggren, Einige Eigenshaften der Einheiten reeller quadratischer und rein-biquadratischer Zahl Körper auf die Lösung einer Klasse unbestimmer Gleichungen 4. Grades, Skr. Norske Vid.-Akad. Oslo I Mat.-Naturv. Klasse 12 (1936), 73 pp.

  9. F. Luca, A. Montejano, L. Szalay and A. Togbé, On the X-coordinates of Pell equations which are Tribonacci numbers, Acta Arith. 179 (2017), 25-35.
    MathSciNet     CrossRef

  10. F. Luca and A. Togbé, On the x-coordinates of Pell equations which are Fibonacci numbers, Math. Scand. 122 (2018), 18-30.
    MathSciNet     CrossRef

  11. D.W. Masser and J.H. Rickert, Simultaneous Pell equations, J. Number Theory 61 (1996), 52-66.
    MathSciNet     CrossRef

  12. E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers, II, Izv. Math. 64 (2000), 1217-1269.
    MathSciNet     CrossRef
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