Repdigits as difference of two Fibonacci or Lucas numbers

P. Ray; K. Bhoi

doi:10.30970/ms.56.2.124-132

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

DOI: https://doi.org/10.30970/ms.56.2.124-132

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

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