A characterization of Fibonacci numbers

Abstract

For the early {\it 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
{\it Pythagorean} system is been put in crisis by their own
discovery of the incommensurability of the {\it side} and {\it
diagonal} of a {\it regular pentagon}. This fundamental historical
discovery, glory of the {\it Pythagorean School}, did however ``{\it
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 ``{\it creative
interpretation}'' of the {\it Pythagorean} research that we present
in this paper: the link between the {\it Pythagorean identity}
$b(b+a)-a^2=0$ concerning the {\it side} $b$ and the {\it diagonal}
$a$ of a {\it regular pentagon} and the {\it Cassini identity}
$F_{i}F_{i+2}-F_{i+1}^2=(-1)^{i}$, concerning three consecutive {\it
Fibonacci numbers}, is very strong. Moreover, the two just mentioned
equations were ``{\it almost simultaneously}'' discovered by the
{\it Pythagorean School} and consequently {\it Fibonacci numbers}
and {\it Cassini identity} are of {\it 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are no historical documents (so rare for that period!)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