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...
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Ivan Franko National University of Lviv
2021-12-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/255 |
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| author | P. Ray K. Bhoi |
| author_facet | P. Ray K. Bhoi |
| author_sort | P. Ray |
| collection | DOAJ |
| description | 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). |
| format | Article |
| id | doaj-art-c25e5ce78cad45d78ae12b1bf23bb08b |
| institution | Matheson Library |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2021-12-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-c25e5ce78cad45d78ae12b1bf23bb08b2025-07-08T09:10:15ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202021-12-0156212413210.30970/ms.56.2.124-132255Repdigits as difference of two Fibonacci or Lucas numbersP. Ray0K. Bhoi1Department of Mathematics, Sambalpur University, Jyoti Vihar, Burla, IndiaDepartment of Mathematics Sambalpur University, Jyoti Vihar, Burla, IndiaIn 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).http://matstud.org.ua/ojs/index.php/matstud/article/view/255fibonacci sequencelucas sequencelinear forms in logarithmsbaker-davenport reduction method |
| spellingShingle | P. Ray K. Bhoi Repdigits as difference of two Fibonacci or Lucas numbers Математичні Студії fibonacci sequence lucas sequence linear forms in logarithms baker-davenport reduction method |
| title | Repdigits as difference of two Fibonacci or Lucas numbers |
| title_full | Repdigits as difference of two Fibonacci or Lucas numbers |
| title_fullStr | Repdigits as difference of two Fibonacci or Lucas numbers |
| title_full_unstemmed | Repdigits as difference of two Fibonacci or Lucas numbers |
| title_short | Repdigits as difference of two Fibonacci or Lucas numbers |
| title_sort | repdigits as difference of two fibonacci or lucas numbers |
| topic | fibonacci sequence lucas sequence linear forms in logarithms baker-davenport reduction method |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/255 |
| work_keys_str_mv | AT pray repdigitsasdifferenceoftwofibonacciorlucasnumbers AT kbhoi repdigitsasdifferenceoftwofibonacciorlucasnumbers |