Repdigits in generalized Pell sequences

Jhon J. Bravo; Jose L. Herrera

Archivum Mathematicum (2020)

  • Volume: 056, Issue: 4, page 249-262
  • ISSN: 0044-8753

Abstract

top
For an integer k ≥ 2 , let ( n ) n be the k - generalized Pell sequence which starts with 0 , ... , 0 , 1 ( k terms) and each term afterwards is given by the linear recurrence n = 2 n - 1 + n - 2 + ⋯ + n - k . In this paper, we find all k -generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence ( P n ( 2 ) ) n .

How to cite

top

Bravo, Jhon J., and Herrera, Jose L.. "Repdigits in generalized Pell sequences." Archivum Mathematicum 056.4 (2020): 249-262. <http://eudml.org/doc/296926>.

@article{Bravo2020,
abstract = {For an integer $k\ge 2$, let $(\{n\})_n$ be the $k-$generalized Pell sequence which starts with $0,\ldots ,0,1$ ($k$ terms) and each term afterwards is given by the linear recurrence $\{n\} = 2\{n-1\}+\{n-2\}+\cdots +\{n-k\}$. In this paper, we find all $k$-generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence $(P_n^\{(2)\})_n$.},
author = {Bravo, Jhon J., Herrera, Jose L.},
journal = {Archivum Mathematicum},
keywords = {generalized Pell numbers; repdigits; linear forms in logarithms; reduction method},
language = {eng},
number = {4},
pages = {249-262},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Repdigits in generalized Pell sequences},
url = {http://eudml.org/doc/296926},
volume = {056},
year = {2020},
}

TY - JOUR
AU - Bravo, Jhon J.
AU - Herrera, Jose L.
TI - Repdigits in generalized Pell sequences
JO - Archivum Mathematicum
PY - 2020
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 056
IS - 4
SP - 249
EP - 262
AB - For an integer $k\ge 2$, let $({n})_n$ be the $k-$generalized Pell sequence which starts with $0,\ldots ,0,1$ ($k$ terms) and each term afterwards is given by the linear recurrence ${n} = 2{n-1}+{n-2}+\cdots +{n-k}$. In this paper, we find all $k$-generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence $(P_n^{(2)})_n$.
LA - eng
KW - generalized Pell numbers; repdigits; linear forms in logarithms; reduction method
UR - http://eudml.org/doc/296926
ER -

References

top
  1. Baker, A., Davenport, H., The equations 3 x 2 - 2 = y 2 and 8 x 2 - 7 = z 2 , Quart. J. Math. Oxford Ser. (2) 20 (1969), 129–137. (1969) MR0248079
  2. Bravo, J.J., Gómez, C.A., Luca, F., 10.18514/MMN.2016.1505, Miskolc Math. Notes 17 (1) (2016), 85–100. (2016) MR3527869DOI10.18514/MMN.2016.1505
  3. Bravo, J.J., Herrera, J.L., Luca, F., 10.21136/MB.2020.0098-19, doi:10.21136/MB.2020.0098-19 on line in Math. Bohem. DOI10.21136/MB.2020.0098-19
  4. Bravo, J.J., Luca, F., 10.5486/PMD.2013.5390, Publ. Math. Debrecen 82 (3–4) (2013), 623–639. (2013) MR3066434DOI10.5486/PMD.2013.5390
  5. Bravo, J.J., Luca, F., 10.1007/s12044-014-0174-7, Proc. Indian Acad. Sci. Math. Sci. 124 (2) (2014), 141–154. (2014) MR3218885DOI10.1007/s12044-014-0174-7
  6. Dujella, A., Pethö, A., A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2) 49 (195) (1998), 291–306. (1998) Zbl0911.11018MR1645552
  7. Faye, B., Luca, F., Pell and Pell-Lucas numbers with only one distinct digits, Ann. of Math. 45 (2015), 55–60. (2015) MR3438812
  8. Kiliç, E., 10.1016/j.ejc.2007.03.004, European J. Combin. 29 (2008), 701–711. (2008) MR2397350DOI10.1016/j.ejc.2007.03.004
  9. Kiliç, E., On the usual Fibonacci and generalized order - k Pell numbers, Ars Combin 109 (2013), 391–403. (2013) MR2426404
  10. Kiliç, E., Taşci, D., 10.11650/twjm/1500404581, Taiwanese J. Math. 10 (6) (2006), 1661–1670. (2006) MR2275152DOI10.11650/twjm/1500404581
  11. Koshy, T., Fibonacci and Lucas Numbers with Applications, Pure and Applied Mathematics, Wiley-Interscience Publications, New York, 2001. (2001) MR1855020
  12. Luca, F., Fibonacci and Lucas numbers with only one distinct digit, Port. Math. 57 (2) (2000), 243–254. (2000) Zbl0958.11007MR1759818
  13. Marques, D., On k -generalized Fibonacci numbers with only one distinct digit, Util. Math. 98 (2015), 23–31. (2015) MR3410879
  14. Matveev, E.M., 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, translation in Izv. Math. 64 (2000), no. 6, 1217–1269. (2000) MR1817252
  15. Normenyo, B., Luca, F., Togbé, A., 10.1007/s10998-018-0247-y, Period. Math. Hungarica 77 (2) (2018), 318–328. (2018) MR3866634DOI10.1007/s10998-018-0247-y
  16. Normenyo, B., Luca, F., Togbé, A., 10.1007/s40590-018-0202-1, Bol. Soc. Mat. Mex. (3) 25 (2) (2019), 249–266. (2019) MR3964309DOI10.1007/s40590-018-0202-1

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.