The Fibonacci version of the Brocard-Ramanujan Diophantine equation (Q555198)

Let \(F_k\) denote the \(k\)-th Fibonacci number. In this note the author considers the Diophantine equation \[ F_n \ldots F_1 + 1 = F_m^2 \] in unknown positive integers \(n,m\), which is the Fibonacci version of the Brocard-Ramanujan Diophantine equation \(n! + 1 = m^2\). He proves the unsolubility of a more general equation, namely, it is proved that the product of \(k\) consecutive Fibonacci numbers \(+ 1\) is never a square of a Fibonacci number. The proof depends on the Primitive Divisor Theorem.