Lucas number#Lucas primes

As with the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediately previous terms, thereby forming a Fibonacci integer sequence. The first two Lucas numbers are $L\_0=2$ and $L\_1=1$, which differs from the first two Fibonacci numbers $F\_0=0$ and $F\_1=1$. Though closely related in definition, Lucas and Fibonacci numbers exhibit distinct properties.

The Lucas numbers may thus be defined as follows:

:$$

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}

(where n belongs to the natural numbers)

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 converges to the golden ratio.

Using $L\_\{n-2\}=L\_\{n\}-L\_\{n-1\}$, one can extend the Lucas numbers to negative integers to obtain a doubly infinite sequence:

:..., −11, 7, −4, 3, −1, 2, 1, 3, 4, 7, 11, ... (terms $L\_n$ for $-5\backslash leq\{\}n\backslash leq5$ are shown).

The formula for terms with negative indices in this sequence is

: $L\_\{-n\}=(-1)^nL\_n.\backslash !$

File:Comparison_Fibonacci_and_Lucas_numbers.svg

The Lucas numbers are related to the Fibonacci numbers by many identities. Among these are the following:

• $L\_n\; =\; F\_\{n-1\}+F\_\{n+1\}\; =\; 2F\_\{n+1\}-F\_n$
• $L\_\{m+n\}\; =\; L\_\{m+1\}F\_\{n\}+L\_mF\_\{n-1\}$
• $F\_\{2n\}\; =\; L\_n\; F\_n$
• $F\_\{n+k\}\; +\; (-1)^k\; F\_\{n-k\}\; =\; L\_k\; F\_n$
• $2F\_\{2n+k\}\; =\; L\_\{n\}\; F\_\{n+k\}\; +\; L\_\{n+k\}\; F\_\{n\}$
• $L\_\{2n\}\; =\; 5\; F\_n^2\; +\; 2(-1)^n\; =\; L\_n^2\; -\; 2(-1)^n$, so $\backslash lim\_\{n\backslash to\backslash infty\}\; \backslash frac\{L\_n\}\{F\_n\}=\backslash sqrt\{5\}$.
• $\backslash vert\; L\_n\; -\; \backslash sqrt\{5\}\; F\_n\; \backslash vert\; =\; \backslash frac\{2\}\{\backslash varphi^n\}\; \backslash to\; 0$
• $L\_\{n+k\}\; -\; (-1)^k\; L\_\{n-k\}\; =\; 5\; F\_n\; F\_k$; in particular, $F\_n\; =\; \{L\_\{n-1\}+L\_\{n+1\}\; \backslash over\; 5\}$, so $5F\_n\; +\; L\_n\; =\; 2L\_\{n+1\}$.

Their closed formula is given as:

:$L\_n\; =\; \backslash varphi^n\; +\; (1-\backslash varphi)^\{n\}\; =\; \backslash varphi^n\; +\; (-\; \backslash varphi)^\{-n\}=\backslash left(\{\; 1+\; \backslash sqrt\{5\}\; \backslash over\; 2\}\backslash right)^n\; +\; \backslash left(\{\; 1-\; \backslash sqrt\{5\}\; \backslash over\; 2\}\backslash right)^n\backslash ,\; ,$

where $\backslash varphi$ is the golden ratio. Alternatively, as for $n>1$ the magnitude of the term $(-\backslash varphi)^\{-n\}$ is less than 1/2, $L\_n$ is the closest integer to $\backslash varphi^n$ or, equivalently, the integer part of $\backslash varphi^n+1/2$, also written as $\backslash lfloor\; \backslash varphi^n+1/2\; \backslash rfloor$.

Combining the above with Binet's formula,

:$F\_n\; =\; \backslash frac\{\backslash varphi^n\; -\; (1-\backslash varphi)^\{n\}\}\{\backslash sqrt\{5\}\}\backslash ,\; ,$

a formula for $\backslash varphi^n$ is obtained:

:$\backslash varphi^n\; =\; \{\{L\_n\; +\; F\_n\; \backslash sqrt\{5\}\}\; \backslash over\; 2\}\backslash ,\; .$

For integers n ≥ 2, we also get:

:$\backslash varphi^n\; =\; L\_n\; -\; (-\; \backslash varphi)^\{-n\}\; =\; L\_n\; -\; (-1)^n\; L\_n^\{-1\}\; -\; L\_n^\{-3\}\; +\; R$

with remainder R satisfying

:.

Many of the Fibonacci identities have parallels in Lucas numbers. For example, the Cassini identity becomes

:$L\_n^2\; -\; L\_\{n-1\}L\_\{n+1\}\; =\; (-1)^\{n\}5$

Also

:$\backslash sum\_\{k=0\}^n\; L\_k\; =\; L\_\{n+2\}\; -\; 1$

:$\backslash sum\_\{k=0\}^n\; L\_k^2\; =\; L\_nL\_\{n+1\}\; +\; 2$

:$2L\_\{n-1\}^2\; +\; L\_n^2\; =\; L\_\{2n+1\}\; +\; 5F\_\{n-2\}^2$

where $\backslash textstyle\; F\_n=\backslash frac\{L\_\{n-1\}+L\_\{n+1\}\}\{5\}$.

:$L\_n^k\; =\; \backslash sum\_\{j=0\}^\{\backslash lfloor\; \backslash frac\{k\}\{2\}\; \backslash rfloor\}\; (-1)^\{nj\}\; \backslash binom\{k\}\{j\}\; L\text{'}\_\{(k-2j)n\}$

where $L\text{'}\_n=L\_n$ except for $L\text{'}\_0=1$.

For example if n is odd, $L\_n^3\; =\; L\text{'}\_\{3n\}-3L\text{'}\_n$ and $L\_n^4\; =\; L\text{'}\_\{4n\}-4L\text{'}\_\{2n\}+6L\text{'}\_0$

Checking, $L\_3=4,\; 4^3=64=76-3(4)$, and $256=322-4(18)+6$

Let

:$\backslash Phi(x)\; =\; 2\; +\; x\; +\; 3x^2\; +\; 4x^3\; +\; \backslash cdots\; =\; \backslash sum\_\{n\; =\; 0\}^\backslash infty\; L\_nx^n$

be the generating function of the Lucas numbers. By a direct computation,

:$\backslash begin\{align\}$

\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}

which can be rearranged as

:$\backslash Phi(x)\; =\; \backslash frac\{2\; -\; x\}\{1\; -\; x\; -\; x^2\}$

$\backslash Phi\backslash !\backslash left(-\backslash frac\{1\}\{x\}\backslash right)$ gives the generating function for the negative indexed Lucas numbers, $\backslash sum\_\{n\; =\; 0\}^\backslash infty\; (-1)^nL\_nx^\{-n\}\; =\; \backslash sum\_\{n\; =\; 0\}^\backslash infty\; L\_\{-n\}x^\{-n\}$, and

:$\backslash Phi\backslash !\backslash left(-\backslash frac\{1\}\{x\}\backslash right)\; =\; \backslash frac\{x\; +\; 2x^2\}\{1\; -\; x\; -\; x^2\}$

$\backslash Phi(x)$ satisfies the functional equation

:$\backslash Phi(x)\; -\; \backslash Phi\backslash !\backslash left(-\backslash frac\{1\}\{x\}\backslash right)\; =\; 2$

As the generating function for the Fibonacci numbers is given by

:$s(x)\; =\; \backslash frac\{x\}\{1\; -\; x\; -\; x^2\}$

we have

:$s(x)\; +\; \backslash Phi(x)\; =\; \backslash frac\{2\}\{1\; -\; x\; -\; x^2\}$

which proves that

:$F\_n\; +\; L\_n\; =\; 2F\_\{n+1\},$

and

:$5s(x)\; +\; \backslash Phi(x)\; =\; \backslash frac2x\backslash Phi(-\backslash frac1x)\; =\; 2\backslash frac\{1\}\{1\; -\; x\; -\; x^2\}\; +\; 4\backslash frac\{x\}\{1\; -\; x\; -\; x^2\}$

proves that

:$5F\_n\; +\; L\_n\; =\; 2L\_\{n+1\}$

The partial fraction decomposition is given by

:$\backslash Phi(x)\; =\; \backslash frac\{1\}\{1\; -\; \backslash phi\; x\}\; +\; \backslash frac\{1\}\{1\; -\; \backslash psi\; x\}$

where $\backslash phi\; =\; \backslash frac\{1\; +\; \backslash sqrt\{5\}\}\{2\}$ is the golden ratio and $\backslash psi\; =\; \backslash frac\{1\; -\; \backslash sqrt\{5\}\}\{2\}$ is its conjugate.

This can be used to prove the generating function, as

:$\backslash sum\_\{n\; =\; 0\}^\backslash infty\; L\_nx^n\; =\; \backslash sum\_\{n\; =\; 0\}^\backslash infty\; (\backslash phi^n\; +\; \backslash psi^n)x^n\; =\; \backslash sum\_\{n\; =\; 0\}^\backslash infty\; \backslash phi^nx^n\; +\; \backslash sum\_\{n\; =\; 0\}^\backslash infty\; \backslash psi^nx^n\; =\; \backslash frac\{1\}\{1\; -\; \backslash phi\; x\}\; +\; \backslash frac\{1\}\{1\; -\; \backslash psi\; x\}\; =\; \backslash Phi(x)$

If $F\_n\backslash geq\; 5$ is a Fibonacci number then no Lucas number is divisible by $F\_n$.

The Lucas numbers satisfy Gauss congruence. This implies that $L\_n$ is congruent to 1 modulo $n$ if $n$ is prime. The composite values of $n$ which satisfy this property are known as Fibonacci pseudoprimes.

$L\_n-L\_\{n-4\}$ is congruent to 0 modulo 5.

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, 3010349, 54018521, 370248451, 6643838879, ... {{OEIS|id=A005479}}.

The indices of these primes are (for example, L₄ = 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}}.

{{as of|2015|09}}, the largest confirmed Lucas prime is L₁₄₈₀₉₁, which has 30950 decimal digits.{{cite web |title=The Top Twenty: Lucas Number |url=https://primes.utm.edu/top20/page.php?id=48 |website=primes.utm.edu |access-date=6 January 2022}} {{as of|2022|08}}, the largest known Lucas probable prime is L_5466311, with 1,142,392 decimal digits.{{cite web |title=Henri & Renaud Lifchitz's PRP Top - Search by form |url=http://www.primenumbers.net/prptop/searchform.php?form=L%28n%29&action=Search |website=www.primenumbers.net |access-date=6 January 2022}}

If L_n is prime then n is 0, prime, or a power of 2.Chris Caldwell, "[http://primes.utm.edu/glossary/page.php?sort=LucasPrime The Prime Glossary: Lucas prime]" from The Prime Pages. L_2^m is prime for m = 1, 2, 3, and 4 and no other known values of m.

In the same way as Fibonacci polynomials are derived from the Fibonacci numbers, the Lucas polynomials $L\_\{n\}(x)$ are a polynomial sequence derived from the Lucas numbers.

Close rational approximations for powers of the golden ratio can be obtained from their continued fractions.

For positive integers n, the continued fractions are:

:$\backslash varphi^\{2n-1\}\; =\; [L\_\{2n-1\};\; L\_\{2n-1\},\; L\_\{2n-1\},\; L\_\{2n-1\},\; \backslash ldots]$

:$\backslash varphi^\{2n\}\; =\; [L\_\{2n\}-1;\; 1,\; L\_\{2n\}-2,\; 1,\; L\_\{2n\}-2,\; 1,\; L\_\{2n\}-2,\; 1,\; \backslash ldots]$.

For example:

:$\backslash varphi^5\; =\; [11;\; 11,\; 11,\; 11,\; \backslash ldots]$

is the limit of

:$\backslash frac\{11\}\{1\},\; \backslash frac\{122\}\{11\},\; \backslash frac\{1353\}\{122\},\; \backslash frac\{15005\}\{1353\},\; \backslash ldots$

with the error in each term being about 1% of the error in the previous term; and

:$\backslash varphi^6\; =\; [18\; -\; 1;\; 1,\; 18\; -\; 2,\; 1,\; 18\; -\; 2,\; 1,\; 18\; -\; 2,\; 1,\; \backslash ldots]\; =\; [17;\; 1,\; 16,\; 1,\; 16,\; 1,\; 16,\; 1,\; \backslash ldots]$

is the limit of

:$\backslash frac\{17\}\{1\},\; \backslash frac\{18\}\{1\},\; \backslash frac\{305\}\{17\},\; \backslash frac\{323\}\{18\},\; \backslash frac\{5473\}\{305\},\; \backslash frac\{5796\}\{323\},\; \backslash frac\{98209\}\{5473\},\; \backslash frac\{104005\}\{5796\},\; \backslash ldots$

with the error in each term being about 0.3% that of the second previous term.

Lucas numbers are the second most common pattern in sunflowers after Fibonacci numbers, when clockwise and counter-clockwise spirals are counted, according to an analysis of 657 sunflowers in 2016.{{Cite journal |last1=Swinton |first1=Jonathan |last2=Ochu |first2=Erinma |last3=null |first3=null |title=Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment |journal=Royal Society Open Science |year=2016 |volume=3 |issue=5 |pages=160091 |doi=10.1098/rsos.160091 |pmc=4892450 |pmid=27293788|bibcode=2016RSOS....360091S }}

• Generalizations of Fibonacci numbers

{{reflist}}

• {{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}}

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