On Fibonacci numbers of the form \(q^ky^k\).
The authors study the equations (1) \(F_n=q^ky^p\) and (2) \(L_n=q^ky^p\), where \(k>0\) and \(p\), \(q\) are primes, and \(F_n\) and \(L_n\) denote the \(n\)th Fibonacci and Lucas numbers, respectively. They prove that: (i) if \(q=2\), then the only solutions of (1) occur for \(n=3, 6, 12\); (ii) If \(q\) is odd and \(n\) is even, then similarly \(n=0,4,12\); (iii) if \(q\) and \(n\) are odd, then necessarily \(q\equiv 1\pmod 4\). Furthermore, either \(q=n=5\), or the Fibonacci entry point of \(q\) must be odd, and (1) has a solution such that \(q\nmid n\). The authors also show that if \(q=2\), then (2) has a solution only when \(n=3\). The authors use the software package MAGMA to get solutions of the Diophantine equation arising in the solution of (2) for \(q=2\) and \(y=3\).
scientific article; zbMATH DE number 2118257