A note on the exponential diophantine equation \(\varvec{(a^{n}-1)(b^{n}-1)=x^{2}}\)

  • Refik KeskinEmail author


In 2002, Luca and Walsh (J. Number Theory 96 (2002) 152–173) solved the diophantine equation for all pairs (ab) such that \(2\le a<b\le 100\) with some exceptions. There are sixty nine exceptions. In this paper, we give some new results concerning the equation \((a^{n}-1)(b^{n}-1)=x^{2}\). It is also proved that this equation has no solutions if ab have opposite parity and \(n>4\) with 2|n. Here, the equation is also solved for the pairs \((a,b)=(2,50),(4,49),(12,45),(13,76),(20,77),(28,49),(45,100)\). Lastly, we show that when b is even, the equation \( (a^{n}-1)(b^{2n}a^{n}-1)=x^{2}\) has no solutions nx.


Pell equation exponential diophantine equation Lucas sequence 

Mathematics Subject Classification

11D61 11D31 11B39 


  1. 1.
    Bennet M A and Skinner C M, Ternary diophantine equation via Galois representations and modular forms, Canad. J. Math. 56 (2004) 23–54MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cohn J H E, The diophantine equation \( (a^{n}-1)(b^{n}-1)=x^{2},\) Period. Math. Hungar. 44 (2002) 169–175MathSciNetCrossRefGoogle Scholar
  3. 3.
    Damir M T, Faye B, Luca F and Tall A, Members of Lucas sequences whose Euler function is a power of 2, Fibonacci Quart. 52 (2014) 3–9MathSciNetzbMATHGoogle Scholar
  4. 4.
    Hajdu L and Szalay L, On the diophantine equations \( (2^{n}-1)(6^{n}-1)=x^{2}\) and \((a^{n}-1)(a^{kn}-1)=x^{2}\), Period. Math. Hungar. 40 (2000) 141–145MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ishii K, On the exponential diophantine equation \( (a^{n}-1)(b^{n}-1)=x^{2},\) Pub. Math. Debrecen 89 (2016) 253–256CrossRefGoogle Scholar
  6. 6.
    Keskin R and Şiar Z, Positive integer solutions of some diophantine equations in terms of integer sequences, Afr. Mat. 30 (2019) 181–184MathSciNetCrossRefGoogle Scholar
  7. 7.
    Lan L and Szalay L, On the exponential diophantine equation \( (a^{n}-1)(b^{n}-1)=x^{2}\), Publ. Math. Debrecen 77 (2010) 1–6MathSciNetzbMATHGoogle Scholar
  8. 8.
    Le M H, A note on the exponential diophantine equation \( (2^{n}-1)(b^{n}-1)=x^{2}\), Publ. Math. Debrecen 74 (2009) 401–403MathSciNetzbMATHGoogle Scholar
  9. 9.
    Li Z-J and Tang M, A remark on a paper of Luca and Walsh, Integers 11 (2011) A40, 6 pp.Google Scholar
  10. 10.
    Luca F, Walsh P G, The product of like-indexed terms in binary recurrences, J. Number Theory 96 (2002) 152–173MathSciNetCrossRefGoogle Scholar
  11. 11.
    Luca F, Effective Methods for Diophantine Equations,
  12. 12.
    Ribenboim P, My Numbers, My Friends (2000) (New York: Springer-Verlag)zbMATHGoogle Scholar
  13. 13.
    Szalay L, On the diophantine equations \((2^{n}-1)(3^{n}-1)\) \( =x^{2}\), Publ. Math. Debrecen 57 (2000) 1–9MathSciNetzbMATHGoogle Scholar
  14. 14.
    Şiar Z and Keskin R, Some new identities concerning generalized Fibonacci and Lucas numbers, Hacet. J. Math. Stat. 42 (2013) 211–222MathSciNetzbMATHGoogle Scholar
  15. 15.
    Tang M, A note on the exponential diophantine equation \((a^{m}-1)(b^{n}-1)=x^{2}\), J. Math. Research and Exposition 31(6) (2011) 1064–1066MathSciNetzbMATHGoogle Scholar
  16. 16.
    van der Waall R W, On the diophantine equation \( x^{2}+x+1=3y^{2},x^{3}-1=2y^{2}\) and \(x^{3}+1=2y^{2},\) Simon Stevin 46 (1972/73) 39–51Google Scholar
  17. 17.
    Walker D T, On the diophantine equation \(mX^{2}-nY^{2}=\pm 1,\) Amer. Math. Montly 74 (1967) 504–513zbMATHGoogle Scholar
  18. 18.
    Walsh P G, On diophantine equations of the form \((x^{n}-1)(y^{m}-1)=z^{2}\), Tatra Math. Publ. 20 (2000) 87–89zbMATHGoogle Scholar
  19. 19.
    Xioyan G, A note on the diophantine equation \( (a^{n}-1)(b^{n}-1)=x^{2},\) Period. Math. Hungar. 66 (2013) 87–93MathSciNetCrossRefGoogle Scholar
  20. 20.
    Yuan P and Zhang Z, On the diophantine equation \((a^{n}-1)(b^{n}-1)=x^{2},\) Publ. Math. Debrecen 80 (2012) 327–331MathSciNetCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Arts and ScienceSakarya UniversitySakaryaTurkey

Personalised recommendations