Ranks and integer points on elliptic curves induced by Fibonacci triples
Abstract
Let Fn and Ln denote the Fibonacci and Lucas numbers, respectively, and consider Ek: y2=(F2kx+1)(F2k+2x+1)(F2k+4x+1). These elliptic curves arise naturally from the regular Diophantine triples {F2k,F2k+2,F2k+4}. For odd k, we exhibit the rational point Qk=( −(Fk−1)/(LkFk+1Fk+2), (F2k+1)/(LkFk+1Fk+2) ). For every odd k≥ 3, this point is independent of the standard point Pk=(0,1); in particular, rankEk(ℚ)≥ 2. Moreover, if k≥ 3 is odd and rankEk(ℚ)=2, then all integer points on Ek are exactly the points arising from the two known solutions of the Hoggatt-Bergum extension problem. By parametrizing the two conics L2−5F2=±4 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 $4$.
Related
- Integer points for iterated Weierstrass elliptic functions
- Methods to Find Integer Points on the Elliptic Curve of Factorizable Polynomial Equation
- Metallic Circles: Concyclic Tangency Points of Ford Circles over k-Fibonacci Quotients
- Torsion points in families of elliptic curves
- Integer points close to a transcendental curve: an algorithmic approach
BibTeX
@misc{dujella2026ranks,
title = {{Ranks and integer points on elliptic curves induced by Fibonacci triples}},
author = {Andrej Dujella},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01789},
eprint = {2609.01789},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01789},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ranks and integer points on elliptic curves induced by Fibonacci triples
AU - Andrej Dujella
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01789
UR - https://arxiv.org/abs/2609.01789
AB - Let F_n and L_n denote the Fibonacci and Lucas numbers, respectively, and consider E_k: y^2=(F_2kx+1)(F_2k+2x+1)(F_2k+4x+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=( −(F_k−1)/(L_kF_k+1F_k+2), (F_2k+1)/(L_kF_k+1F_k+2) ). For every odd k≥ 3, this point is independent of the standard point P_k=(0,1); in particular, rankE_k(ℚ)≥ 2. Moreover, if k≥ 3 is odd and rankE_k(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=±4 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 $4$.
ER -