Integer points on the elliptic curve with Fibonacci numbers (Q6572369)

The Fibonacci sequence \( (F_n)_{n\ge 0} \) is a well-known example of a second-order recurrence sequence, defined by the recurrence relation \( F_0=0 \), \( F_1=1 \), and \( F_n=F_{n-1}+F_{n-2} \) for all \( n\ge 2 \). A set \( \{a_1,a_2,\ldots, a_m\} \) of positive integers is called a Diophantine \( m \)-tuple if \( a_ia_j+1 \) is a perfect square for all \( 1\le i<j\le m \). For instance, Fermat first found the integer set \(\{1, 3, 8, 120\}\) which satisfies the property that the product of any two of them is one less than a perfect square. In the paper under review, the author finds the structure of the torsion group of the elliptic curve \( E_k \) induced by the Diophantine triple \( \{ F_{2K}, F_{2k+4}, 5F_{2k+2}\} \) constructed by Fibonacci numbers, and find all integer points on \( E_k \) under the assumption that rank\((E_k(\mathbb{Q}))=1\).