Leírás
Online Number Theory Seminar
Abstract: Assume that $k\ge2$ is a given positive integer. The $k$-generalized Lucas sequence $\{L_{n}^{(k)}\}_{n\ge0}$ has positive integer initial values $k$, $1$, $3$, $\ldots$, $2^{k-1}-1$, and each term
afterward is the sum of the $k$ consecutive preceding elements:
$$
L_{n}^{(k)}=L_{n-1}^{(k)}+L_{n-2}^{(k)}+\cdots+L_{n-k}^{(k)}.
$$
An integer $n$ is said to be close to a positive integer $m$ if it satisfies $|n-m|<\sqrt{m}$.
In the talk, we solve completely a closeness problem, namely the diophantine inequality
\begin{equation*}
\left| L_{n}^{(k)}-2^{m}\right|<2^{m/2}
\end{equation*}
in the non-negative integers $k$, $n$, and $m$. This problem is equivalent to the resolution of the equation $L_{n}^{(k)}=2^{m}+t$
with the condition $\left\vert t\right\vert <2^{m/2}$, $t\in\mathbb{Z}$. We also discovered a new formula for $L_{n}^{(k)}$ which was very useful in the investigation of one particular case of the problem.
For access please contact the organizers (ntrg[at]science.unideb.hu).