论文标题
在$ k $ generalized fibonacci数字带有负指数的数字
On the $k$-generalized Fibonacci numbers with negative indices
论文作者
论文摘要
在这些注释中，我们研究了$ k $ generalized fibonacci序列 - $（f_n^{（k）}）_ {n \ in \ z} $，带有正和负数。表示$ t_k（x）$其特征多项式。我们最有趣的发现是，如果$ k $甚至是$ t_k（x）$的第二个真实根的绝对值在根中是最小的。将其与Bugeaud和Kaneko \ cite {bk}的深层结果相结合，我们证明$（f_n^{（k）}）_ {n \ in \ z} $，提供$ k $偶数。另一个结果是，如果$ k $和$ l $表示甚至整数，则方程$ f_m^{（k）} = \ pm f_n^{（l）} $在$（n，m）中只有有效地计算的解决方案有限地有效地计算解决方案。在情况下，$ k = l = 4 $，我们建立了该方程的所有解决方案。
In these notes we study the $k$-generalized Fibonacci sequences - $(F_n^{(k)})_{n\in \Z}$ - with positive and negative indices. Denote $T_k(x)$ its characteristic polynomial. Our most interesting finding is that if $k$ is even then the absolute value of the second real root of $T_k(x)$ is minimal among the roots. Combining this with a deep result of Bugeaud and Kaneko \cite{BK} we prove that there are only finitely many perfect powers in $(F_n^{(k)})_{n\in \Z}$, provided $k$ is even. Another consequence is that, if $k$ and $l$ denote even integers then the equation $F_m^{(k)} = \pm F_n^{(l)}$ has only finitely many effectively computable solutions in $(n,m)\in \Z^2$. In the case $k=l=4$ we establish all solutions of this equation.
