Ranks and integer points on elliptic curves induced by Fibonacci triples

Abstract

Let $F_n$ and $L_n$ denote the Fibonacci and Lucas numbers, respectively, and consider \[ E_k:\qquad y^2=(F_{2k}x+1)(F_{2k+2}x+1)(F_{2k+4}x+1). \] These elliptic curves arise naturally from the regular Diophantine triples \[ \{F_{2k},F_{2k+2},F_{2k+4}\}. \] For odd $k$, we exhibit the rational point \[ Q_k=\left( -\frac{F_{k-1}}{L_kF_{k+1}F_{k+2}}, \frac{F_{2k+1}}{L_kF_{k+1}F_{k+2}} \right). \] For every odd $k\geq 3$, this point is independent of the standard point $P_k=(0,1)$; in particular, $\operatorname{rank}E_k(\mathbb{Q})\geq 2$. Moreover, if $k\geq 3$ is odd and $\operatorname{rank}E_k(\mathbb{Q})=2$, then all integer points on $E_k$ are exactly the points arising from the two known solutions of the Hoggatt-Bergum extension problem. By parametrizing the two conics $L^2-5F^2=\pm4$ and applying an injective specialization criterion, we also show that the corresponding one-parameter elliptic families have generic ranks $2$ in the odd case and $1$ in the even case. Finally, we discuss computational data and propose the heuristic rank distribution $1/4,1/2,1/4$ for ranks $1,2,3$, respectively, with density zero for rank at least $4$.