Repdigits as difference of two Fibonacci or Lucas numbers
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 $(n,m)\in \{(7,3),(9,1),(9,2),(11,1),(11,2),$ $(11,9),(12,11),(15,10)\}.$ Furt...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | German |
| Published: |
Ivan Franko National University of Lviv
2021-12-01
|
| Series: | Математичні Студії |
| Subjects: | |
| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/255 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | 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). |
|---|---|
| ISSN: | 1027-4634 2411-0620 |