# 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) |

**How to Cite This Entry:**

Fibonacci numbers.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Fibonacci_numbers&oldid=12076