A characterization of Fibonacci numbers

The link between the equation $b(b+a)-a^2=0$ concerning the side $b$ and the diagonal $a$ of a regular pentagon and the {\it Cassini identity} $F_{i}F_{i+2}-F_{i+1}^2=(-1)^{i}$, concerning three consecutive Fibonacci numbers, is very strong. In this paper we present our thesis that the two mentioned equations were "almost simultaneously" discovered by the {\it Pythagorean School}.