## Abstract

Let q be a perfect power of a prime number p and E(F_{q}) be an elliptic curve over F_{q} given by the equation y^{2}=x^{3}+Ax+B. For a positive integer n we denote by #E(F_{qn}) the number of rational points on E (including infinity) over the extension F_{qn}. Under a mild technical condition, we show that the sequence {#E(F_{qn})}_{n>0} contains at most 10^{200} perfect squares. If the mild condition is not satisfied, then #E(F_{qn}) is a perfect square for infinitely many n including all the multiples of 12. Our proof uses a quantitative version of the Subspace Theorem. We also find all the perfect squares for all such sequences in the range q<50 and n≤1000.

Original language | English |
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Article number | 101725 |

Journal | Finite fields and their applications |

Volume | 67 |

DOIs | |

Publication status | Published - Oct 2020 |

## Keywords

- Elliptic curves
- Recurrence sequence
- Subspace theorem

## ASJC Scopus subject areas

- Theoretical Computer Science
- Algebra and Number Theory
- Engineering(all)
- Applied Mathematics