| Title:
|
On the intersection of two distinct $k$-generalized Fibonacci sequences (English) |
| Author:
|
Marques, Diego |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
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2464-7136 (online) |
| Volume:
|
137 |
| Issue:
|
4 |
| Year:
|
2012 |
| Pages:
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403-413 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $k\geq 2$ and define $F^{(k)}:=(F_n^{(k)})_{n\geq 0}$, the $k$-generalized Fibonacci sequence whose terms satisfy the recurrence relation $F_n^{(k)}=F_{n-1}^{(k)}+F_{n-2}^{(k)}+\cdots + F_{n-k}^{(k)}$, with initial conditions $0,0,\dots ,0,1$ ($k$ terms) and such that the first nonzero term is $F_1^{(k)}=1$. The sequences $F:=F^{(2)}$ and $T:=F^{(3)}$ are the known Fibonacci and Tribonacci sequences, respectively. In 2005, Noe and Post made a conjecture related to the possible solutions of the Diophantine equation $F_n^{(k)}=F_m^{(\ell )}$. In this note, we use transcendental tools to provide a general method for finding the intersections $F^{(k)}\cap F^{(m)}$ which gives evidence supporting the Noe-Post conjecture. In particular, we prove that $F\cap T=\{0,1,2,13\}$. (English) |
| Keyword:
|
$k$-generalized Fibonacci numbers |
| Keyword:
|
linear forms in logarithms |
| Keyword:
|
reduction method |
| MSC:
|
11B39 |
| MSC:
|
11D61 |
| MSC:
|
11J86 |
| idZBL:
|
Zbl 1258.11026 |
| idMR:
|
MR3058272 |
| DOI:
|
10.21136/MB.2012.142996 |
| . |
| Date available:
|
2012-11-10T20:29:08Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142996 |
| . |
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