LUCAS NUMBER

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The 'Lucas numbers' are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers (both are Lucas sequences). Much like the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediate previous terms. Consequently, the ratio between two consecutive Lucas numbers converges to the golden ratio.
However, the first two Lucas numbers are ''L''₀ = 2 and ''L''₁ = 1 instead of 0 and 1, and the properties of Lucas numbers are therefore somewhat different from those of Fibonacci numbers.
A Lucas number may thus be defined as follows:
:$$
L_n := L(n):=
egin{cases}
2 & mbox{if } n = 0; \
1 & mbox{if } n = 1; \
L(n-1)+L(n-2) & mbox{if } n > 1. \
end{cases}

The sequence of Lucas numbers begins:
:2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ...

Contents

Extension to negative integers

Relationship to Fibonacci numbers

Congruence Relation

Lucas primes

External links

Extension to negative integers

Using L_n-2 = L_n - L_n-1, one can extend the Lucas numbers to negative integers. So we get: ... -11, 7, -4, 3, -1, 2, 1, 3, 4, 7, 11, ... .
:$L\_\{-n\}=(-1)^nL\_n.!$

Relationship to Fibonacci numbers

The Lucas numbers are related to the Fibonacci numbers by the identities

★ $,L\_n\; =\; F\_\{n-1\}+F\_\{n+1\}$

★ $,F\_\{2n\}\; =\; L\_n\; F\_n$
Their closed formula is given as:
:
where is the Golden ratio.
Also:

★ $,F\_n\; =\; \{L\_\{n-1\}+L\_\{n+1\}\; over\; 5\}$
As $n,$ approaches infinity $L\_n\; over\; F\_n,$ approaches $sqrt\{5\},\; .$

Congruence Relation

L_n is congruent to 1 mod n if n is prime, but some composite values of n also have this property.

Lucas primes

A 'Lucas prime' is a Lucas number that is prime. The first few Lucas primes are
2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, ...
Except for the cases ''n'' = 0, 4, 8, 16, if ''L_n'' is prime then ''n'' is prime. The converse is false, however.
See also Fibonacci prime.

External links

★ MathWorld

★ Dr Ron Knott

★ Lucas numbers and the Golden Section

★ A Lucas Number Calculator can be found here.