# Repdigits as difference of two Fibonacci or Lucas numbers

• P. Ray Department of Mathematics, Sambalpur University, Jyoti Vihar, Burla, India
• K. Bhoi Department of Mathematics Sambalpur University, Jyoti Vihar, Burla, India
Keywords: Fibonacci sequence, Lucas sequence, linear forms in logarithms, Baker-Davenport reduction method

### Abstract

In the present study we investigate all repdigits which are expressed as a difference of two Fibonacci or Lucas numbers. We show that if $F_{n}-F_{m}$ is a repdigit, where $F_{n}$ denotes the $n$-th Fibonacci number, then $(n,m)\in \{(7,3),(9,1),(9,2),(11,1),(11,2),$ $(11,9),(12,11),(15,10)\}.$ Further, if $L_{n}$ denotes the $n$-th Lucas number, then $L_{n}-L_{m}$ is a repdigit for $(n,m)\in\{(6,4),(7,4),(7,6),(8,2)\},$ where $n>m.$
Namely, the only repdigits that can be expressed as difference of two Fibonacci numbers are $11,33,55,88$ and $555$; their representations are $11=F_{7}-F_{3},\ 33=F_{9}-F_{1}=F_{9}-F_{2},\ 55=F_{11}-F_{9}=F_{12}-F_{11},\ 88=F_{11}-F_{1}=F_{11}-F_{2},\ 555=F_{15}-F_{10}$ (Theorem 2). Similar result for difference of two Lucas numbers: The only repdigits that can be expressed as difference of two Lucas numbers are $11,22$ and $44;$ their representations are $11=L_{6}-L_{4}=L_{7}-L_{6},\ 22=L_{7}-L_{4},\ 4=L_{8}-L_{2}$ (Theorem 3).

### Author Biographies

P. Ray, Department of Mathematics, Sambalpur University, Jyoti Vihar, Burla, India

Department of Mathematics

Sambalpur University, Jyoti Vihar, Burla, India

K. Bhoi, Department of Mathematics Sambalpur University, Jyoti Vihar, Burla, India

Department of Mathematics

Sambalpur University, Jyoti Vihar, Burla, India

### References

C. Adegbindin, F. Luca, A. Togb´e, Lucas numbers as sums of two repdigits, Lith. Math. J., 59 (2019), 295–304.

A. Alahmadi, A. Altassan, F. Luca, H. Shoaib, Fibonacci numbers which are concatenation of two repdigits, Quaest. Math., (2019), https://doi.org/10.2989/16073606.2019.1686439.

Y. Bugeaud, M. Mignotte, S. Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Ann. of Math. (2), 163 (2006), 969–1018.

H. Cohen, A course in computational algebraic number thoery, Springer, New York, 1993.

H. Cohen, Number theory. Volume I: Tools and Diophantine equations, Springer, New York, 2007.

A. Dujella, A. Peth˝o, A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser., 49 (1998), 291–306.

F. Erduvan, R. Keskin, F. Luca, Fibonacci and Lucas numbers as difference of two repdigits, Rend. Circ. Mat. Palermo, II. Ser, (2021) https://doi.org/10.1007/s12215-021-00645-3.

S. G´uzman S´anchez, F. Luca, Linear combinations of factorials and s-units in a binary recurrence sequence, Ann. Math. du Qu´e, 38 (2014), 169–188.

F. Luca, Fibonacci and Lucas numbers with only one distinct digit, Port. Math., 57 (2000), 243–254.

D. Marques, A. Togb´e, On repdigits as product of consecutive Fibonacci numbers, Rend. Istit. Mat. Univ. Trieste, 44 (2012), 393–397.

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 (2000), 125–180. Translation in Izv. Math. 64 (2000), 1217–1269.

Published
2021-12-26
How to Cite
Ray, P., & Bhoi, K. (2021). Repdigits as difference of two Fibonacci or Lucas numbers. Matematychni Studii, 56(2), 124-132. https://doi.org/10.30970/ms.56.2.124-132
Issue
Section
Articles