Positive integer solutions of certain diophantine equations

  • Bijan Kumar Patel
  • Prasanta Kumar Ray
  • Manasi K Sahukar


In this study, the diophantine equations \(x^2 -32B_nxy-32y^2 =\pm 32^{r}\), \(x^4 -32B_nxy-32y^2 =\pm 32^{r}\) and \(x^2 -32B_nxy-32y^4 =\pm 32^{r}\) are considered and determined when these equations have positive integer solutions. Moreover, all positive integer solutions of these diophantine equations in terms of balancing and Lucas-balancing numbers are also found out.


Diophantine equations balancing numbers balancers Lucas balancing numbers 

2010 Mathematics Subject Classification

11B37 11B39 11D45 


  1. 1.
    Behera A and Panda G K, On the square roots of triangular numbers, The Fibonacci Quart.  37(2) (1999) 98–105MathSciNetzbMATHGoogle Scholar
  2. 2.
    Cohn J H E, The Diophantine equation \(x^4- D y^2= 1\): II, Acta Arith.  78(4) (1997) 401–403CrossRefGoogle Scholar
  3. 3.
    Cohn J H E, Perfect Pell powers, Glasg. Math. J.  38 (1996) 19–20MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cusick T W, The Diophantine equation \(x^4- kx^2y^2+ y^4= 1\), Arch. Math.  59(4) (1992) 345–347MathSciNetCrossRefGoogle Scholar
  5. 5.
    Karaatli O, Keskin R and Zhu H, Infinitely many positive integer solutions of the quadratic diophantine equations \(x^2-8B_nxy-2y^2= \pm 2^r,\) Irish Math. Soc. Bull.  73 (2014) 29–45zbMATHGoogle Scholar
  6. 6.
    Keskin R, Karaatli O and Şiar Z, Positive integer solutions of the Diophantine equations \(x^{2} -5F_nxy-5(-1)^ny^2=\pm 5^r\), Miskolc Math. Notes  14(3) (2013) 959–972MathSciNetzbMATHGoogle Scholar
  7. 7.
    Keskin R and Yosma Z, Positive integer solutions of the Diophantine equations \(x^{2} - L_{n}xy + (-1)^ny^2=\pm 5^{r}\), Proc. Indian Acad. Sci. (Math. Sci.)  124(3) (2014) 301–313Google Scholar
  8. 8.
    Luo J-G and Yuan P-Z, On square-classes in Lucas sequences, Adv. Math. (Chinese)  35(2) (2006) 211–216MathSciNetGoogle Scholar
  9. 9.
    McDaniel W L, The G.C.D. in Lucas sequences and Lehmer number sequences, Fibonacci Quart.  29(1) (1991) 24–29MathSciNetzbMATHGoogle Scholar
  10. 10.
    Nakamula K and Petho A, Squares in binary recurrence sequences, in: Number Theory, edited by K. Gyory et al., de Gruyter, Berlin (1998) 409–421Google Scholar
  11. 11.
    Panda G K and Ray P K, Some links of balancing and cobalancing numbers with Pell and associated Pell numbers, Bull. Inst. Math. Acad. Sin. (New Series)  6(1) (2011) 41–72MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ray P K, Some congruences for balancing and Lucas-balancing numbers and their applications, Integers  14 (2014) #A8.Google Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  • Bijan Kumar Patel
    • 1
  • Prasanta Kumar Ray
    • 2
  • Manasi K Sahukar
    • 3
  1. 1.International Institute of Information TechnologyBhubaneswarIndia
  2. 2.VSS University of Technology, OdishaBurlaIndia
  3. 3.National Institute of TechnologyRourkelaIndia

Personalised recommendations