Elliptic curves over finite fields with Fibonacci numbers of points (Q782954)

The paper studies and solves a curious problem: let \(E\)\, be an elliptic curve defined over a finite field \(\mathbb{F}_q\)\, with \(\sharp(E)=q+1-a\)\, and let us denote \(E_m(q,a);\,\, m\ge 1\)\, the number of points of \(E\)\, over \(\mathbb{F}_{q^m}\); let \(\{F_n\}_{n\ge 1}\)\, be the Fibonacci sequence. When the intersection of \(\{E_m(q,a)\}_{m\ge 1}\)\, and \(\{F_n\}_{n\ge 1}\)\, has two elements at least?, i.e. to find the solutions of \(\sharp E_{m_1} = F_{n_1}\),\, \(\sharp E_{m_2} = F_{n_2}\), etc. Section 1 discusses the problem and formulates the obtained result (Theorem 1.1): the cardinal of the intersection is two for four couples \((q,a)\)\, and three in only a case (for \(q=a=2\)). The rest of the paper is devoted to the proof of that theorem. Section 2 studies the equation \(E_m(q,a)=F_n\)\, in terms of linear forms in logarithms and discusses how the problem can be reduced to three cases: (i) \(q\)\, small (\(q\le 10.000\)), (ii) \(n_1\)\, small and (iii) \(m_2\)\, small. This provides a list of the possible values for \(q\). Section 3 gathers some necessary tools regarding linear forms in logarithms and continued fractions. Assuming proved that \(n_2\le 1.000\)\, Section 4 first studies the problem for \(q\le 10.000\)\, and, using the computational package \texttt{Mathematica} finds the five solutions of Theorem 1.1. Then, for \(q> 10.000\),\, also using Mathematica, proves that there is no solution. The rest of the paper assumes \(n_2> 1.000\)\, and Section 5 to 10 deals with the three cases (i), (ii) and (iii) and, always with the computational help of Mathematica (the paper says that ``the total calculation time for the Mathematica software for this paper was 20 days on 25 parallel desktop computers'') finishes the proof of Theorem 1.1.