호남수학학술지 (Honam Mathematical Journal)

  • 제46권4호
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  • Pages.552-566
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  • 2024
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  • 1225-293X(pISSN)
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  • 2288-6176(eISSN)
호남수학회 (The Honam Mathematical Society)
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PERFECT POWERS AS DIFFERENCE OF PERRIN NUMBERS AND PADOVAN NUMBERS

  • Merve Guney Duman (Fundamental Science in Engineering, Sakarya University of Applied Sciences)
  • 투고 : 2023.10.22
  • 심사 : 2024.09.11
  • 발행 : 2024.12.20
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초록

In this paper, we investigate the perfect powers that are the difference between Perrin numbers (Rk)k≥0 and Padovan numbers (Pk)k≥0. Hence, we solve the equations Pn = xa, 2Pn = xa, Pn - Rm = xa, or Rn - Pm = xa such that a ≥ 1 and 2 ≤ x ≤ 10 are positive integers and n, m, and k are non-negative integers.

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과제정보

I would like to thank the referees for helpful suggestions.

참고문헌

  1. A. Baker and H. Davenport, The equations 3x2 - 2 = y2 and 8x2 - 7 = z2, Q. J. Math. Oxford Series (2) 20 (1969), no. 1, 129-137.
  2. W. D. Banks and F. Luca, Concatenations with binary recurrent sequences, J. Integer Seq. 8 (2005). (Article 05.1.3).
  3. D. Bitim and R. Keskin, On solutions of the Diophantine equation Fn - Fm = 3a, Proceedings- Mathematical Sciences 129 (2019), no. 81.
  4. J. J. Bravo and F. Luca, On the Diophantine equation Fn + Fm = 2a, Quaest. Math. 39 (2016), no. 3, 391-400.
  5. J. J. Bravo, C.A. Gomez, and F. Luca, Powers of two as sums of two k-Fibonacci numbers, Miskolc Math Notes 17 (2016), no. 1, 85-100.
  6. Y. Bugeaud, M. Mignotte, and S. Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Ann. Math. 163 (2006), no. 3, 969-1018.
  7. Y. Bugeaud, F. Luca, M. Mignotte, and S. Siksek, Fibonacci numbers at most one away from a perfect power, Elem. Math. 63 (2008), 65-75.
  8. Y. Bugeaud, Linear forms in logarithms and applications, IRMA Lectures in Mathematics and Theoretical Physics, 28, Zurich: European Mathematical Society, 2018.
  9. A. Dujella and A. Petho, A generalization of a theorem of Baker and Davenport, Q. J. Math. Oxford Series 49 (1998), no. 3, 291-306.
  10. F. Erduvan and R. Keskin, Non-negative integer solutions of the Diophantine equation Fn - Fm = 5a, Turk. J. Math. 43 (2019), 115-123.
  11. D. G. M. Farrokhi, Some remarks on the equation Fn = kFm in Fibonacci numbers, J. Integer Seq. 10 (2007), no. 2, (Article 07.5.7)
  12. M. Guney Duman, U. Ogut, and R. Keskin, Generalized Lucas numbers of the form wx2 and wVmx2, Hokkaido Math. J. 47 (2018), no. 3, 465-480.
  13. O. Karaatli and R. Keskin, On the Lucas sequence equations Vn = 7x2 and Vn = 7Vmx2, Bull. Malays. Math. Sci. Soc. 41 (2018), 335-353.
  14. R. Keskin and B. Demirturk, Fibonacci and Lucas congruence a their applications, Acta Mathematica Sinica (English Ser.) 27 (2011), 725-736.
  15. R. Keskin and Z. Siar, On the Lucas sequence equations Vn = kVm and Un = kUm, Colloq. Math. 130 (2013), 27-38.
  16. A. M. Legendre, Essai sur la theorie des nombres, Duprat, Paris, 1798.
  17. F. Luca and V. Patel, On perfect powers that are sums of two Fibonacci numbers, J. Number Theory 189 (2018), 90-96.
  18. E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II, Izvestiya Akademii Nauk Series Mathematics 64 (2000), no. 6, 125-180 (in Russian).
  19. S.C. Patel, S.G. Rayaguru, P. Tiebekabe, G.K. Panda, and K.R. Kakanou, On solutions of the Diophantine equation Ln ± Lm = pa, Malaya Journal of Matematik 11 (2023), no. 3, 294-302.
  20. P. Ribenboim and W.L. McDaniel, Squares in Lucas sequences having an even first parameter, Colloq. Math. 78 (1998), 29-34.
  21. P. Ribenboim and W.L. McDaniel, On Lucas sequence terms of the form kx2, Number theory: Proceedings of the Turku Symposium on Number Theory in Memory of Kustaa Inkeri (Turku, 1999), 293-303, Walter de Gruyter, Berlin, 2001.
  22. S. E. Rihane and A. Togbe, On the intersection of Padovan, Perrin sequences and Pell, Pell-Lucas sequences, Ann. math. inform. 54 (2021), 57-71.
  23. Z. Siar and R. Keskin, On the Diophantine equation Fn - Fm = 2a, Colloq. Math. 159 (2020), 119-126.
  24. P. Tiebekabe and I. Diouf, On solutions of the Diophantine equation Ln + Lm = 3a, Malaya Journal of Matematik 9 (2021), no. 4, 228-238.
  25. P. Tiebekabe and I. Diouf, Power of three as difference of two Fibonacci numbers, JP J Algebr. Number Theory 49 (2021), no. 2, 185-196.
  26. B. M. M. De Weger, Algorithms for Diophantine equations, CWI Tracts 65, Stichting Mathematisch Centrum, Amsterdam, 1989.