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).

Keywords

• fibonacci sequence
• lucas sequence
• linear forms in logarithms
• baker-davenport reduction method