In: Tatra Mountains Mathematical Publications, vol. 67, no. 3
Rok, strany: 2016, 41 - 46
$k$-Fibonacci number, $k$-Lucas number, Galois theory, Diophantine equation
Let $k≥ 1$ and denote $(Fk,n)n≥ 0$, the $k$-Fibonacci sequence whose terms satisfy the recurrence relation $Fk,n=kFk,n-1+Fk,n-2$, with initial con di tions $Fk,0=0$ and $Fk,1=1$. In the same way, the $k$-Lucas sequence $(Lk,n)n≥ 0$ is defined by satisfying the same recurrence relation with initial values $Lk,0=2$ and $Lk,1=k$. These sequences were introduced by Falcon and Plaza, who showed many of their properties, too. In particular, they proved that $Fk,n+1+Fk,n-1=Lk,n$, for all $k≥ 1$ and $n≥ 0$. In this paper, we shall prove that if $k≥1$ and $Fk,n+1s+Fk,n-1s\in (Lk,m)m≥ 1$ for infinitely many positive integers $n$, then $s=1$.
Trojovský, P. 2016. On the sum of powers of two $k$-Fibonacci numbers which belongs to the sequence of $k$-Lucas numbers. In Tatra Mountains Mathematical Publications, vol. 67, no.3, pp. 41-46. 1210-3195.
Trojovský, P. (2016). On the sum of powers of two $k$-Fibonacci numbers which belongs to the sequence of $k$-Lucas numbers. Tatra Mountains Mathematical Publications, 67(3), 41-46. 1210-3195.