The elements of the sequence
given by the initial values
and the recurrence relation
.
The first 14 Fibonacci numbers were produced for the
first time in
1228
in the manuscripts of
Leonardo da Pisa
(Fibonacci).
Operations that can be performed on the indices of the Fibonacci numbers
can be reduced to operations on the numbers themselves.
The basis for this lies in the
"addition formula" :
Immediate corollaries of it are:
etc. The general
"multiplication formula"
is more complicated:
The elementary divisibility properties of the Fibonacci numbers
are mainly determined by the following facts:
;
if
is a prime number of the form
,
then
is divisible by
,
while if it is of the form
,
then
is; if
is divisible by a prime number
and if
,
then
is not divisible by
;
if
is divisible by a prime number
,
then
is divisible by
,
but not by
;
if
is divisible by 4, then
is divisible by 2, but not by 4; if
is divisible by 2 but not by 4, then
is divisible by 4, but not by 8. At
the same time, some number-theoretic problems connected with Fibonacci numbers are
extremely hard. For example, the question of whether the set of prime
Fibonacci numbers is finite or not has not been solved
(1984).
An important role in the theory of Fibonacci numbers is played by the number
,
which is a root of the equation
.
Thus
Binet's formula
holds; it implies that
is the nearest integer to
,
and that
The Fibonacci numbers occupy a special position in the theory of
continued fractions.
In the continued-fraction expansion of
all the partial quotients are ones and the number of them is not
less than that of the incomplete quotients of the
expansion of any other fraction with denominator less than
.
In a certain sense the number
is described by its approximating fractions
in a
"worst possible"
way.