Korean J. Math. Vol. 31 No. 1 (2023) pp.63-73
DOI: https://doi.org/10.11568/kjm.2023.31.1.63

Repdigits as difference of two Pell or Pell-Lucas numbers

Main Article Content

Refik Keskin
Fatih Erduvan

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 $444=478-34=Q_{7}-Q_{4}.$



Article Details

References

[1] A. Baker and H. Davenport, The equations 3x²-2=y² and 8x²-7=z², Quart. J. Math. Oxford Ser. 2, 20 (1) (1969),129-137. https://doi.org/10.1093/qmath/20.1.129 Google Scholar

[2] J.J. Bravo, C.A. Gomez and F.Luca, Powers of two as sums of two k-Fibonacci numbers, Miskolc Math. Notes. 17 (1) (2016), 85–100. Google Scholar

[3] Y. Bugeaud, Linear Forms in Logarithms and Applications, IRMA Lectures in Mathematics and Theoretical Physics, 28, Zurich: European Mathematical Society, 2018. Google Scholar

[4] Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Ann. of Math. 163 (3) (2006), 969– 1018. Google Scholar

[5] A. Dujella and A. Peth`o, A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2), 49 (3) (1998), 291–306. Google Scholar

[6] B. Faye and F. Luca, Pell and Pell-Lucas numbers with only one distinct digit, Ann. Math. Inform. 45 (2015), 55–60. Google Scholar

[7] F. Luca, Fibonacci and Lucas numbers with only one distinct digit, Portugal. Math. 57 (2) (2000), 243–254. Google Scholar

[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) (2000), 125–180 (Russian). Translation in Izv. Math. 64 (6) (2000), 1217–1269. Google Scholar

[9] B.V. Normenyo, F. Luca and A. Togb ́e, Repdigits as sums of three Pell numbers, Periodica Mathematica Hungarica. 77 (2) (2018), 318–328. Google Scholar

[10] S. G. Rayaguru and G. K. Panda, Repdigits As Sums Of Two Associated Pell Numbers, Applied Mathematics E-Notes. 21 (2021), 402–409. Google Scholar

[11] Z. S ̧iar, F. Erduvan and R. Keskin, Repdigits as Products of two Pell or Pell-Lucas Numbers, Acta Mathematica Universitatis Comenianae. 88 (2) (2019), 247–256. Google Scholar

[12] Z. S ̧iar, F. Erduvan and R. Keskin, Repdigits base b as difference of two Fibonacci numbers, J. Math. Study. 55 (1) (2022), 84–94. Google Scholar

[13] B. M. M. de Weger, Algorithms for Diophantine Equations, CWI Tracts 65, Stichting Mathematisch Centrum, Amsterdam,1989. https://www.win.tue.nl/~bdeweger/downloads/CWI%20Tract%2065.pdf Google Scholar