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{{distinguish|text=[[Lucas sequence]]s, the generic class of sequences to which the Lucas numbers belong}}
[[File:Lucas number spiral.svg|400x240px|thumb|right|The Lucas spiral, made with quarter-arcs, is a good approximation of the [[golden spiral]] when its terms are large. However, when its terms become very small, the arc's radius decreases rapidly from 3 to 1 then increases from 1 to 2.]]
The '''Lucas numbers''' or '''Lucas series''' are an [[integer sequence]] named after the mathematician [[Édouard Lucas|François Édouard Anatole Lucas]] (1842&ndash;91), who studied both that sequence and the closely related [[Fibonacci number]]s. Lucas numbers and Fibonacci numbers form complementary instances of [[Lucas sequence]]s.

The Lucas sequence has the same recursive relationship as the [[Fibonacci sequence]], where each term is the sum of the two previous terms, but with different starting values. This produces a sequence where the ratios of successive terms approach the [[golden ratio]], and in fact the terms themselves are [[rounding]]s of integer powers of the golden ratio.<ref>{{cite book |last1=Parker |first1=Matt |title=Things to Make and Do in the Fourth Dimension |date=2014 |publisher=Farrar, Straus and Giroux |isbn=978-0-374-53563-6 |page=284 |language=English |chapter=13}}</ref> The sequence also has a variety of relationships with the Fibonacci numbers, like the fact that adding any two Fibonacci numbers two terms apart in the Fibonacci sequence results in the Lucas number in between.<ref>{{cite book |last1=Parker |first1=Matt |title=Things to Make and Do in the Fourth Dimension |date=2014 |publisher=Farrar, Straus and Giroux |isbn=978-0-374-53563-6 |page=282 |language=English |chapter=13}}</ref>

== Definition ==
Similar to the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediate previous terms, thereby forming a [[Generalizations of Fibonacci numbers#Fibonacci integer sequences|Fibonacci integer sequence]].  The first two Lucas numbers are ''L''<sub>0</sub> = 2 and ''L''<sub>1</sub> = 1 as opposed to the first two Fibonacci numbers ''F''<sub>0</sub> = 0 and ''F''<sub>1</sub> = 1.  Though closely related in definition, Lucas and Fibonacci numbers exhibit distinct properties.

The Lucas numbers may thus be defined as follows:
:<math> 
  L_n :=
  \begin{cases}
    2               & \text{if } n = 0; \\
    1               & \text{if } n = 1; \\
    L_{n-1}+L_{n-2} & \text{if } n > 1. \\
   \end{cases}
 </math>
(where n belongs to the natural numbers)

The sequence of Lucas numbers is:
:<math>2,\;1,\;3,\;4,\;7,\;11,\;18,\;29,\;47,\;76,\;123,\; \ldots\;</math>{{OEIS|id=A000032}}.

All Fibonacci-like integer sequences appear in shifted form as a row of the [[Wythoff array]]; the Fibonacci sequence itself is the first row and the Lucas sequence is the second row. Also like all Fibonacci-like integer sequences, the ratio between two consecutive Lucas numbers [[limit of a sequence|converges]] to the [[golden ratio]].

==Extension to negative integers==
Using ''L''<sub>''n''&minus;2</sub> = ''L''<sub>''n''</sub>&nbsp;&minus;&nbsp;''L''<sub>''n''&minus;1</sub>, one can extend the Lucas numbers to negative integers to obtain a doubly infinite sequence: 
:..., &minus;11, 7, &minus;4, 3, &minus;1, 2, 1, 3, 4, 7, 11, ...  (terms <math>L_n</math> for <math>-5\leq{}n\leq5</math> are shown).
The formula for terms with negative indices in this sequence is
: <math>L_{-n}=(-1)^nL_n.\!</math>

==Relationship to Fibonacci numbers==
[[File:Comparison_Fibonacci_and_Lucas_numbers.svg|thumb|300px|The first identity expressed visually]]
The Lucas numbers are related to the Fibonacci numbers by many identities.  Among these are the following:
* <math>L_n = F_{n-1}+F_{n+1}=F_n+2F_{n-1} = F_{n+2}-F_{n-2}</math>
* <math>L_{m+n} = L_{m+1}F_{n}+L_mF_{n-1}</math>
* <math>L_n^2 = 5 F_n^2 + 4 (-1)^n</math>, and thus as <math>n\,</math> approaches [[Extended real number line|+∞]], the ratio <math>\frac{L_n}{F_n}</math> approaches <math>\sqrt{5}.</math>
* <math>F_{2n} = L_n F_n</math>
* <math>F_{n+k} + (-1)^k F_{n-k} = L_k F_n</math>
* <math>L_{n+k} - (-1)^k L_{n-k} = 5 F_n F_k</math>; in particular, <math>F_n = {L_{n-1}+L_{n+1} \over 5}</math>

Their [[Closed-form expression|closed formula]] is given as:
:<math>L_n = \varphi^n + (1-\varphi)^{n} = \varphi^n + (- \varphi)^{-n}=\left({ 1+ \sqrt{5} \over 2}\right)^n + \left({ 1- \sqrt{5} \over 2}\right)^n\, ,</math>

where <math>\varphi</math> is the [[golden ratio]]. Alternatively, as for <math>n>1</math> the magnitude of the term <math>(-\varphi)^{-n}</math> is less than 1/2, <math>L_n</math> is the closest integer to <math>\varphi^n</math> or, equivalently, the integer part of <math>\varphi^n+1/2</math>, also written as <math>\lfloor \varphi^n+1/2 \rfloor</math>.

Combining the above with [[Binet's formula]],

:<math>F_n = \frac{\varphi^n - (1-\varphi)^{n}}{\sqrt{5}}\, ,</math>

a formula for <math>\varphi^n</math> is obtained:

:<math>\varphi^n = {{L_n + F_n \sqrt{5}} \over 2}\, .</math>

==Congruence relations==
If ''F''<sub>''n''</sub> &ge; 5 is a Fibonacci number then no Lucas number is divisible by ''F''<sub>''n''</sub>.

''L''<sub>''n''</sub> is congruent to 1 mod&nbsp;''n'' if ''n'' is prime, but some composite values of ''n'' also have this property. These are the [[Lucas pseudoprime#Fibonacci pseudoprimes|Fibonacci pseudoprimes]].

''L''<sub>''n''</sub> - ''L''<sub>''n-4''</sub> is congruent to ''0'' mod ''5''.

== Lucas primes ==
A '''Lucas prime''' is a Lucas number that is [[prime number|prime]]. The first few Lucas primes are
:2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, ... {{OEIS|id=A005479}}.

The indices of these primes are (for example, ''L''<sub>4</sub> = 7)
:0, 2, 4, 5, 7, 8, 11, 13, 16, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113, 313, 353, 503, 613, 617, 863, 1097, 1361, 4787, 4793, 5851, 7741, 8467, ... {{OEIS|id=A001606}}.

If ''L<sub>n</sub>'' is prime then ''n'' is either 0, prime, or a power of 2.<ref>Chris Caldwell, "[http://primes.utm.edu/glossary/page.php?sort=LucasPrime The Prime Glossary: Lucas prime]" from The [[Prime Pages]].</ref> ''L''<sub>2<sup>''m''</sup></sub> is prime for ''m''&nbsp;= 1,&nbsp;2,&nbsp;3,&nbsp;and&nbsp;4 and no other known values of&nbsp;''m''.

==Generating series==
Let

:<math>\Phi(x) = 2 + x + 3x^2 + 4x^3 + \cdots = \sum_{n = 0}^\infty L_nx^n</math>

be the [[generating series]] of the Lucas numbers. By a direct computation,

:<math>\begin{align}
  \Phi(x) &= L_0 + L_1x + \sum_{n = 2}^\infty L_nx^n \\
          &= 2 + x + \sum_{n = 2}^\infty (L_{n - 1} + L_{n - 2})x^n \\
          &= 2 + x + \sum_{n = 1}^\infty L_nx^{n + 1} + \sum_{n = 0}^\infty L_nx^{n + 2} \\
          &= 2 + x + x(\Phi(x) - 2) + x^2 \Phi(x)
\end{align}</math>

which can be rearranged as

:<math>\Phi(x) = \frac{2 - x}{1 - x - x^2}.</math>


The partial fraction decomposition is given by

:<math>\Phi(x) = \frac{1}{1 - \varphi x} + \frac{1}{1 - \phi x}</math>

where <math>\varphi = \frac{1 + \sqrt{5}}{2}</math> is the golden ratio and <math>\phi = \frac{1 - \sqrt{5}}{2}</math> is its conjugate.

==Lucas polynomials==
In the same way as [[Fibonacci polynomial]]s are derived from the [[Fibonacci number]]s, the [[Lucas polynomials]] ''L''<sub>''n''</sub>(''x'') are a [[polynomial sequence]] derived from the Lucas numbers.

==See also==
*[[Generalizations of Fibonacci numbers]]

== References ==
{{reflist}}

==External links==
* {{springer|title=Lucas polynomials|id=p/l130120}}
* {{MathWorld|urlname=LucasNumber|title=Lucas Number}}
* {{MathWorld | urlname=LucasPolynomial | title=Lucas Polynomial}}
* "[https://web.archive.org/web/20051126021243/http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/lucasNbs.html The Lucas Numbers]", Dr Ron Knott
* [https://web.archive.org/web/20051030021553/http://milan.milanovic.org/math/english/lucas/lucas.html Lucas numbers and the Golden Section]
* [https://web.archive.org/web/20070216024906/http://www.plenilune.pwp.blueyonder.co.uk/fibonacci-calculator.asp A Lucas Number Calculator can be found here.]
* {{OEIS el|1=A000032|2=Lucas numbers beginning at 2|formalname=Lucas numbers beginning at 2: L(n) = L(n-1) + L(n-2), L(0) = 2, L(1) = 1}}

{{Prime number classes}}
{{Classes of natural numbers}}

[[Category:Integer sequences]]
[[Category:Fibonacci numbers]]
[[Category:Recurrence relations]]
[[Category:Unsolved problems in mathematics]]

[[bn:লুকাস ধারা]]
[[de:Lucas-Folge]]
[[fr:Suite de Lucas]]
[[he:סדרת לוקאס]]
[[pt:Sequência de Lucas]]