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Repdigits as difference of two Pell or Pell-Lucas numbers

Refik Keskin|Fatih Erduvan
강원경기수학회 (The Kangweon-Kyungki Mathematical Society)|2023.03|한국수학논문집 (Korean Journal of Mathematics)|제31권 제1호|pp. 63-73

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참고문헌

키워드

Diophantine equationsPell numbersPell-Lucas numbersRepdigitlinear forms in logarithms.
주제분류수학

초록

In this paper, we determine all repdigits, which are difference of two Pell and Pell-Lucas numbers. It is shown that the largest repdigit which is difference of two Pell numbers is $99=169-70=P_{7}-P_{6}$ and the largest repdigit which is difference of two Pell-Lucas numbers is $444=478-34=Q_{7}-Q_{4}.$

Abstract

In this paper, we determine all repdigits, which are difference of two Pell and Pell-Lucas numbers. It is shown that the largest repdigit which is difference of two Pell numbers is $99=169-70=P_{7}-P_{6}$ and the largest repdigit which is difference of two Pell-Lucas numbers is $444=478-34=Q_{7}-Q_{4}.$

참고문헌 (13)

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  2. J. J. Bravo, "Powers of two as sums of two k-Fibonacci numbers", Miskolc Math. Notes, 17(1), pp.85~100 (2016)
  3. Y. Bugeaud, "Linear Forms in Logarithms and Applications" (2018), European Mathematical Society
  4. Y. Bugeaud, "Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers", Ann. of Math, 163(3), pp.969~1018 (2006)
  5. A. Dujella, "A generalization of a theorem of Baker and Davenport", Quart. J. Math. Oxford Ser, 49(3), pp.291~306 (1998)
  6. B. Faye, "Pell and Pell-Lucas numbers with only one distinct digit", Ann. Math. Inform, 45, pp.55~60 (2015)
  7. F. Luca, "Fibonacci and Lucas numbers with only one distinct digit", Portugal. Math, 57(2), pp.243~254 (2000)
  8. E. M. Matveev, "An Explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II", Izv. Ross. Akad. Nauk Ser. Mat, 64(6), pp.1217~1269 (2000)
  9. B. V. Normenyo, "Repdigits as sums of three Pell numbers", Periodica Mathematica Hungarica, 77(2), pp.318~328 (2018)
  10. S. G. Rayaguru, "Repdigits As Sums Of Two Associated Pell Numbers", Applied Mathematics E-Notes, 21, pp.402~409 (2021)
  11. Z. S¸iar, "Repdigits as Products of two Pell or Pell-Lucas Numbers", Acta Mathematica Universitatis Comenianae, 88(2), pp.247~256 (2019)
  12. Z. S¸iar, "Repdigits base b as difference of two Fibonacci numbers", J. Math. Study, 55(1), pp.84~94 (2022)
  13. B. M. M. de Weger, "Algorithms for Diophantine Equations", CWI Tracts, 65 (1989), Stichting Mathematisch Centrum, Amsterdam

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