# Repdigits as difference of two Fibonacci or Lucas numbers

### 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).

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*Matematychni Studii*,

*56*(2), 124-132. https://doi.org/10.30970/ms.56.2.124-132

Copyright (c) 2021 P. Ray, K. Bhoi

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Matematychni Studii is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license.