×

Repdigits in the base \(b\) as sums of four balancing numbers. (English) Zbl 1495.11010

Summary: The sequence of balancing numbers \((B_{n})\) is defined by the recurrence relation \(B_{n}=6B_{n-1}-B_{n-2}\) for \(n\geq 2\) with initial conditions \(B_{0}=0\) and \(B_{1}=1\). \(B_{n}\) is called the \(n\)th balancing number. In this paper, we find all repdigits in the base \(b\), which are sums of four balancing numbers. As a result of our theorem, we state that if \(B_{n}\) is repdigit in the base \(b\) and has at least two digits, then \((n,b)=(2,5),(3,6)\). Namely, \(B_{2}=6=(11)_{5}\) and \(B_{3}=35=(55)_{6}\).

MSC:

11A63 Radix representation; digital problems
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11J86 Linear forms in logarithms; Baker’s method
11D61 Exponential Diophantine equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Weger, B. M. M. de, Algorithms for Diophantine Equations, CWI Tract 65. Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam (1989) · Zbl 0687.10013
[2] Alvarado, S. Díaz; Luca, F., Fibonacci numbers which are sums of two repdigits, Proc. 14th Int. Conf. Fibonacci Numbers and their Applications. Morelia, 2010 Aportaciones Mat. Investig. 20. Soc. Mat. Mexicana, México (2011), 97-108 F. Luca et al · Zbl 1287.11021
[3] Faye, B.; Luca, F., Pell and Pell-Lucas numbers with only one distinct digit, Ann. Math. Inform. 45 (2015), 55-60 · Zbl 1349.11023
[4] Luca, F., Fibonacci and Lucas numbers with only one distinct digit, Port. Math. 57 (2000), 243-254 · Zbl 0958.11007
[5] Luca, F., Repdigits as sums of three Fibonacci numbers, Math. Commun. 17 (2012), 1-11 · Zbl 1305.11008
[6] Luca, F.; Normenyo, B. V.; Togbe, A., Repdigits as sums of four Pell numbers, Bol. Soc. Mat. Mex., III. Ser. 25 (2019), 249-266 · Zbl 1455.11019 · doi:10.1007/s40590-018-0202-1
[7] Keskin, R.; Karaatlı, O., Some new properties of balancing numbers and square triangular numbers, J. Integer Seq. 15 (2012), Article 12.1.4, 13 pages · Zbl 1291.11030
[8] Matveev, E. M., An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II, Izv. Math. 64 (2000), 1217-1269 translation from Izv. Ross. Akad. Nauk, Ser. Mat. 64 2000 125-180 · Zbl 1013.11043 · doi:10.1070/IM2000v064n06ABEH000314
[9] Normenyo, B. V.; Luca, F.; Togbé, A., Repdigits as sums of three Pell numbers, Period. Math. Hung. 77 (2018), 318-328 · Zbl 1413.11008 · doi:10.1007/s10998-018-0247-y
[10] Panda, G. K., Some fascinating properties of balancing numbers, Cong. Numerantium 194 (2009), 185-189 · Zbl 1262.11019
[11] Panda, G. K.; Ray, P. K., Cobalancing numbers and cobalancers, Int. J. Math. Math. Sci. 2005 (2005), 1189-1200 · Zbl 1085.11017 · doi:10.1155/IJMMS.2005.1189
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.