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2 papers
A characterization of Fibonacci numbers
G. Pirillo Dipartimento di Matematica ed Informatica “U. Dini”, Università di Firenze, Florence, Italy
Abstract:
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are no historical documents (so rare for that period!)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.
Keywords:
incommensurability, golden ratio, Fibonacci numbers.
UDC:
510 Received: 11.06.2018
Accepted: 17.08.2018
Language: English
DOI:
10.22405/2226-8383-2018-19-2-259-271
Fulltext:
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