Fibonacci numbers
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.
References
[1] | B. Boncompagni, "Illiber Abbaci di Leonardo Pisano" , Rome (1857) |
[2] | N.N. Vorob'ev, "Fibonacci numbers" , Moscow (1984) (In Russian) |
[3] | V.E. Hoggatt, "Fibonacci and Lucas numbers" , Univ. Santa Clara (1969) |
[4] | U. (or A. Brousseau) Alfred, "An introduction to Fibonacci discovery" , San José, CA (1965) |
[5] | Fibonacci Quart. (1963-) |
Comments
Let ,
be non-zero integers with
and
. A Lucas sequence, or a sequence of Lucas numbers, is defined by
and the linear recurrence relation
![]() |
Still more generally, a sequence of complex numbers, i.e. a number-theoretic or arithmetic function, is said to be recurrent of order
if there is a complex-valued function
of
variables such that
,
. If
is linear,
is called a linear recurrent sequence. Both the Fibonacci and the Lucas numbers are linearly recurrent of order 2.
For some more results on Fibonacci numbers, Lucas numbers and recurrent sequences, as well as for their manifold applications, cf. also [a1].
References
[a1] | A.N. Phillipou (ed.) G.E. Bergum (ed.) A.F. Horodam (ed.) , Fibonacci numbers and their applications , Reidel (1986) |
Fibonacci numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fibonacci_numbers&oldid=12076