On the exponential Diophantine equation \(F_{n+1}^x - F_{n-1}^x = F_m\) (Q2050151)

Let \((F_n)_{n\ge 0}\) be the Fibonacci sequence given by \(F_0=0,~F_1=1\) and \(F_{n+2}=F_{n+1}+F_n\) for all \(n\ge 0\). The authors look at the Diophantine equation in the title in nonnegative integers \((m,n,x)\). The above equation has two parametric families of solutions with \(x=1\) and \(x=2\) because of the identities \(F_{n+1}-F_{n-1}=F_n\) and \(F_{n+1}^2-F_{n-1}^2=F_{2n}\). The authors prove that there are no other solutions \((m,n,x)\) with \(x\ge 3\) and \(n\ge 3\). The proof uses linear forms in logarithms of algebraic numbers and reduction methods based on continued fractions.