A curious property of the eleventh Fibonacci number (Q1912628)

Let \(F_n\) be the \(n\)th term of the Fibonacci sequence, defined by \(F_0 =0\), \(F_1=1\), \(F_{n+2}= F_{n+1}+ F_n\). The author shows that the largest solution of the diophantine equation \(F_n= y^2- y-1\) is \(F_{11}= 89\). This shows that the largest base \(b\) for which \(1/F_n= \sum^\infty_{k=0} F_k/ y^{k+1}\) is \(y=10\). The proof is accomplished by reducing the problem to two quartic Thue equations and then using a result of \textit{A. Baker} and \textit{G. Wüstholz} [J. Reine Angew. Math. 442, 19-62 (1993; Zbl 0788.11026)] and diophantine approximation to solve them.