A characterization of Fibonacci numbers

For the early Pythagoreans, in perfect agreement with their philosophical-mathematical thought, given segments $s$ and $t$ there was a segment $u$ contained exactly $n$ times in $s$ and $m$ times in $t$, for some suitable integers $n$ and $m$. In the sequel, the Pythagorean system is been put in crisis by their own discovery of the incommensurability of the side and diagonal of a regular pentagon. This fundamental historical discovery, glory of the Pythagorean School, did however “ forget” the research phase that preceded their achievement. This phase, started with numerous attempts, all failed, to find the desired common measure and culminated with the very famous odd even argument, is precisely the object of our “creative interpretation” of the Pythagorean research that we present in this paper: the link between the Pythagorean identity $b(b+a)-a^2=0$ concerning the side $b$ and the diagonal $a$ of a regular pentagon and the Cassini identity $F_{i}F_{i+2}-F_{i+1}^2=(-1)^{i}$, concerning three consecutive Fibonacci numbers, is very strong. Moreover, the two just mentioned equations were “almost simultaneously” discovered by the Pythagorean School and consequently Fibonacci numbers and Cassini identity are of Pythagorean origin. There are no historical documents (so rare for that period!) concerning our audacious thesis, but we present solid mathematical arguments that support it. These arguments provide in any case a new (and natural!) characterization of the Fibonacci numbers, until now absent in literature.