View More View Less
  • 1 Department of Mathematics, Bursa Uludağ University, 16059 Bursa, Turkey
  • 2 Institute of Mathematics, Lingnan Normal College, Zhanjiang, 524048 China
  • 3 Department of Mathematics, Bursa Uludağ University, 16059 Bursa, Turkey
Restricted access

Purchase article

USD  $25.00

1 year subscription (Individual Only)

USD  $800.00

Abstract

Let l,m,r be fixed positive integers such that 2| l, 3lm, l > r and 3 | r. In this paper, using the BHV theorem on the existence of primitive divisors of Lehmer numbers, we prove that if min{rlm2 − 1,(lr)lm2 + 1} > 30, then the equation (rlm2 − 1)x + ((lr)lm2 + 1)y = (lm)z has only the positive integer solution (x,y,z) = (1,1,2).

  • [1]

    Bertók, Cs., The complete solution of the Diophantine equation, Period. Math. Hung., 72 (2016), 3742.

  • [2]

    Bilu, Y., Hanrot, G. and Voutier, P. M., Existence of primitive divisors of Lucas and Lehmer numbers, with an appendix by M. Mignotte, J. Reine Angew. Math., 539 (2001), 75122.

    • Search Google Scholar
    • Export Citation
  • [3]

    Fu, R., Yang, H., On the exponential Diophantine equation (am 2 + 1)x + (bm 2 − 1)y = (cm) z with c | m, Period. Math. Hung., 75 (2017), 143149.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [4]

    Kızıldere, E., Miyazaki, T. and Soydan, G., On the Diophantine equation ((c+1)m 2 +1)x +(cm 2 −1)y = (am) z, Turkish J. of Math., 42 (2018), 26902698.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [5]

    Le, M.-H., Some exponential Diophantine equations I: The equation D 1 x 2D 2 y 2 = λk z, J. of Number Theory, 55 (1995), 209221.

  • [6]

    Miyazaki, T. and Terai, N., On the exponential Diophantine equation (m 2 + 1)x + (cm 2 − 1)y = (am) z, Bull. Aust. Math. Soc., 90 (2014), 919.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [7]

    Pan, X., A note on the exponential Diophantine equation (am 2 + 1)x + (bm 2 − 1)y = (cm) z, Colloq. Math., 149 (2017), 265273.

  • [8]

    Soydan, G., Demirci, M., Cangül, I. N. and Togbé, A., On the conjecture of Jesmanowicz, Int. J. of Appl. Math. and Stat., 56 (2017), 4672.

    • Search Google Scholar
    • Export Citation
  • [9]

    Su, J. and Li, X., The exponential Diophantine equation (4m 2+1)x+(5m 2−1)y = (3m) z, Abstr. Appl. Anal., 2014 (2014), 15.

  • [10]

    Terai, N., On the exponential Diophantine equation (4m 2 + 1)x + (5m 2 − 1)y = (3m) z, Int. J. of Algebra, 6 (2012), 11351146.

  • [11]

    Terai, N. and Hibino, T., On the exponential Diophantine equation (12m 2 +1)x + (13m 2 − 1)y = (5m) z, Int. J. of Algebra, 9 (2015), 261272.

  • [12]

    Terai, N. and Hibino, T., On the exponential Diophantine equation (3pm 2 −1)x + (p(p − 3)m 2 + 1)y = (pm) z, Period. Math. Hung., 74-2 (2017), 227234.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [13]

    Wang, J.-P., Wang, T.-T. and Zhang, W.-P., A note on the exponential Diophantine equation (4m 2 + 1)x + (5m 2 − 1)y = (3m)z, Colloq. Math., 139 (2015), 121126.

    • Crossref
    • Search Google Scholar
    • Export Citation