New York Journal of Mathematics
Volume 26 (2020), 711-734


Yuri Bilu, Carlos A. Gómez, Jhonny C. Gómez, and Florian Luca

Elliptic curves over finite fields with Fibonacci numbers of points

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Published: June 23, 2020.
Keywords: Fibonacci numbers; elliptic curves; linear forms in logarithms; Baker-Davenport reduction.
Subject: 11D61; 11G20, 11J86, 11Y50, 11B39.

For a prime power q and an elliptic curve E over Fq having q+1-a points, where a ∈ [-2√q,2√q], let #Em, m ≥ 1, be the sequence of numbers whose mth term is the number of points of E over Fqm. In this paper, we determine all instances when #({#Em}∩{F_n})≥ 2, where {Fn} (n≥1) is the sequence of Fibonacci numbers. That is, we determine all six--tuples (a,q,m1,m2,n1,n2) such that #E=q+1-a, #Em1=Fn1 and #Em2=Fn2.


The authors thank the Department of Mathematics at the Universidad del Valle for the Cluster time used to perform calculations, and especially the Computer Center Jurgen Tischer for their advice on parallelisation of the algorithm used. Yu. B. worked on this paper when he was visiting Universidad de Valparaíso, supported by the Project MEC 80160038, and Institute of Mathematical Sciences in Chennai, supported by SPARC Project P445. He thanks Instituto de Matemáticas UV and IMSc for excellent working conditions. C.A.G. was supported in part by Project 71228 (Universidad del Valle). J.C.G. worked on this project during a visit to the FLAME in Morelia, Mexico, in August 2019. This author also thanks the Universidad del Valle for support during his Ph.D. studies. F.L worked on this paper while he was visiting the Max Planck Institute for Mathematics in Fall 2019. He thanks this institution for hospitality and support. In addition, F. L. was also supported in part by grant CPRR160325161141 from the NRF of South Africa and the Focus Area Number Theory grant RTNUM19 from CoEMaSS Wits.

The authors thank Karim Belabas, Yann Bugeaud and Amalia Pizarro-Madariaga for stimulating discussions and helpful suggestions, and the referee for careful reading of the manuscript and useful comments.

Author information

Yuri Bilu:
IMB, Université de Bordeaux & CNRS
351 cours de la Libération
33405, Talence CEDEX, France


Carlos A. Gómez:
Departamento de Matemáticas
Universidad del Valle
25360 Cali, Calle 13 No 100-00, Colombia


Jhonny C. Gómez:
Departamento de Matemáticas
Universidad del Valle
25360 Cali, Calle 13 No 100-00, Colombia


Florian Luca:
School of Mathematics
Wits University
Johannesburg, South Africa;
Research Group in Algebraic Structures and Applications
King Abdulaziz University
Jeddah, Saudi Arabia;
Centro de Ciencias Matemáticas
Morelia, Mexico