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Repdigits as sums of two Fibonacci numbers and two Lucas numbers.(English)Zbl 1467.11009

Summary: In this paper, we completely determine all repdigits in base 10 which can be expressed as sums of two Fibonacci numbers and two Lucas numbers.

MSC:

11A63 Radix representation; digital problems
11B39 Fibonacci and Lucas numbers and polynomials and generalizations

Software:

OEIS

Cite

Online Encyclopedia of Integer Sequences:

Repdigit numbers, or numbers whose digits are all equal.

References:

[1]D. W. Ballew, and R. C. Weger, Repdigit triangular numbers, J. Recreational Math. 8 (197576), 96-98. ·Zbl 0327.10014
[2]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 (2006), 969-1018. ·Zbl 1113.11021
[3]A. Dujella and A. Peth˝o, A generalization of a theorem of Baker and Davenport, Quart. J. Math. 49 (1998), 291-306. ·Zbl 0911.11018
[4]M. Keith, Repdigit polygonal numbers, J. Recreational Math. 12 (1979-80), 9-15. ·Zbl 0413.10006
[5]F. Luca, Distinct digits in base b expansions of linear recurrences, Quaest. Math. 23 (2000), 389-404. ·Zbl 1030.11004
[6]F. Luca, Repdigits as sums of three Fibonacci numbers, Math. Commun. 17 (2012), 1-11. ·Zbl 1305.11008
[7]F. Luca, B. V. Normenyo, and A. Togb´e, Repdigits as sums of four Pell numbers, Bol. Soc. Mat. Mex. 25 (2019), 249-266. ·Zbl 1455.11019
[8]F. Luca, B. V. Normenyo, and A. Togb´e, Repdigits as sums of three Lucas numbers, Colloq. Math. 156 (2019), 255-265. ·Zbl 1459.11041
[9]D. Marques, and A. Togb´e, On repdigits as product of consecutive Fibonacci numbers, Rend. Istit. Mat. Univ. Trieste 44 (2012), 393-397. ·Zbl 1290.11008
[10]E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers, II, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), 125-180 (English translation Izv. Math. 64 (2000), 1217-1269). ·Zbl 1013.11043
[11]B. V. Normenyo, F. Luca, and A. Togb´e, Repdigits as sums of four Fibonacci or Lucas numbers, J. Integer Seq. 21 (2018), Article 18.7.7. ·Zbl 1453.11006
[12]B. V. Normenyo, F. Luca, and A. Togb´e, Repdigits as sums of three Pell numbers, Period. Math. Hungar. 77 (2018), 318-328. ·Zbl 1413.11008
[13]N. J. A. Sloane, The Online Encyclopedia of Integer Sequences, published electronically at https://oeis.org. ·Zbl 1044.11108
[14]B. M. M. de Weger, Algorithms for Diophantine Equations, Stichting Mathematisch Centrum, Amsterdam, 1989. ·Zbl 0687.10013
[15]M. Waldshmidt, Diophantine Approximation on Linear Algebraic Groups : Transcendence Properties of the Exponential Function in Several Variables, Springer-Verlag Berlin Heidelberg, Berlin, 2000. ·Zbl 0944.11024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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