Lucas number

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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 sequence is an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci sequence. Individual numbers in the Lucas sequence are known as Lucas numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences.

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.<ref name="mathworld wolfram weisstein">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> This produces a sequence where the ratios of successive terms approach the golden ratio, and in fact the terms themselves are roundings of integer powers of the golden ratio.<ref>Template:Cite book</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>Template:Cite book</ref>

The first few Lucas numbers are

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, ... . (sequence A000032 in the OEIS)

which coincides for example with the number of independent vertex sets for cyclic graphs <math>C_n</math> of length <math>n\geq2</math>.<ref name="mathworld wolfram weisstein" />

DefinitionEdit

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 <math>L_0=2</math> and <math>L_1=1</math>, which differs from the first two Fibonacci numbers <math>F_0=0</math> and <math>F_1=1</math>. 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)

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.

Extension to negative integersEdit

Using <math>L_{n-2}=L_{n}-L_{n-1}</math>, 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 <math>L_n</math> for <math>-5\leq{}n\leq5</math> are shown).

The formula for terms with negative indices in this sequence is

Relationship to Fibonacci numbersEdit

File:Comparison Fibonacci and Lucas numbers.svg

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} = 2F_{n+1}-F_n</math>
<math>L_{m+n} = L_{m+1}F_{n}+L_mF_{n-1}</math>
<math>F_{2n} = L_n F_n</math>
<math>F_{n+k} + (-1)^k F_{n-k} = L_k F_n</math>
<math>2F_{2n+k} = L_{n} F_{n+k} + L_{n+k} F_{n}</math>
<math>L_{2n} = 5 F_n^2 + 2(-1)^n = L_n^2 - 2(-1)^n</math>, so <math>\lim_{n\to\infty} \frac{L_n}{F_n}=\sqrt{5}</math>.
<math> \vert L_n - \sqrt{5} F_n \vert = \frac{2}{\varphi^n} \to 0 </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>, so <math>5F_n + L_n = 2L_{n+1}</math>.

Their 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>

For integers n ≥ 2, we also get:

<math> \varphi^n = L_n - (- \varphi)^{-n} = L_n - (-1)^n L_n^{-1} - L_n^{-3} + R </math>

with remainder R satisfying

<math> \vert R \vert < 3 L_n^{-5} </math>.

Lucas identitiesEdit

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

Also

where <math>\textstyle F_n=\frac{L_{n-1}+L_{n+1}}{5}</math>.

<math> L_n^k = \sum_{j=0}^{\lfloor \frac{k}{2} \rfloor} (-1)^{nj} \binom{k}{j} L'_{(k-2j)n} </math>

where <math>L'_n=L_n</math> except for <math>L'_0=1</math>.

For example if n is odd, <math>L_n^3 = L'_{3n}-3L'_n</math> and <math>L_n^4 = L'_{4n}-4L'_{2n}+6L'_0</math>

Checking, <math>L_3=4, 4^3=64=76-3(4)</math>, and <math>256=322-4(18)+6</math>

Generating functionEdit

Let

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

be the generating function 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\!\left(-\frac{1}{x}\right)</math> gives the generating function for the negative indexed Lucas numbers, <math>\sum_{n = 0}^\infty (-1)^nL_nx^{-n} = \sum_{n = 0}^\infty L_{-n}x^{-n}</math>, and

<math>\Phi\!\left(-\frac{1}{x}\right) = \frac{x + 2x^2}{1 - x - x^2}</math>

<math>\Phi(x)</math> satisfies the functional equation

<math>\Phi(x) - \Phi\!\left(-\frac{1}{x}\right) = 2</math>

As the generating function for the Fibonacci numbers is given by

we have

which proves that

and

proves that

The partial fraction decomposition is given by

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

This can be used to prove the generating function, as

<math>\sum_{n = 0}^\infty L_nx^n = \sum_{n = 0}^\infty (\phi^n + \psi^n)x^n = \sum_{n = 0}^\infty \phi^nx^n + \sum_{n = 0}^\infty \psi^nx^n = \frac{1}{1 - \phi x} + \frac{1}{1 - \psi x} = \Phi(x)</math>

Congruence relationsEdit

If <math>F_n\geq 5</math> is a Fibonacci number then no Lucas number is divisible by <math>F_n</math>.

The Lucas numbers satisfy Gauss congruence. This implies that <math>L_n</math> is congruent to 1 modulo <math>n</math> if <math>n</math> is prime. The composite values of <math>n</math> which satisfy this property are known as Fibonacci pseudoprimes.

<math>L_n-L_{n-4}</math> is congruent to 0 modulo 5.

Lucas primesEdit

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, ... (sequence A005479 in the OEIS).

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, ... (sequence A001606 in the OEIS).

Template:As of, the largest confirmed Lucas prime is L₁₄₈₀₉₁, which has 30950 decimal digits.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Template:As of, the largest known Lucas probable prime is L_5466311, with 1,142,392 decimal digits.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

If L_n is prime then n is 0, prime, or a power of 2.<ref>Chris Caldwell, "The Prime Glossary: Lucas prime" from The Prime Pages.</ref> L_2^m is prime for m = 1, 2, 3, and 4 and no other known values of m.

Lucas polynomialsEdit

In the same way as Fibonacci polynomials are derived from the Fibonacci numbers, the Lucas polynomials <math>L_{n}(x)</math> are a polynomial sequence derived from the Lucas numbers.

Continued fractions for powers of the golden ratioEdit

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

For positive integers n, the continued fractions are:

<math> \varphi^{2n-1} = [L_{2n-1}; L_{2n-1}, L_{2n-1}, L_{2n-1}, \ldots] </math>

<math> \varphi^{2n} = [L_{2n}-1; 1, L_{2n}-2, 1, L_{2n}-2, 1, L_{2n}-2, 1, \ldots] </math>.

For example:

<math> \varphi^5 = [11; 11, 11, 11, \ldots] </math>

is the limit of

<math> \frac{11}{1}, \frac{122}{11}, \frac{1353}{122}, \frac{15005}{1353}, \ldots </math>

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

<math> \varphi^6 = [18 - 1; 1, 18 - 2, 1, 18 - 2, 1, 18 - 2, 1, \ldots] = [17; 1, 16, 1, 16, 1, 16, 1, \ldots] </math>

is the limit of

<math> \frac{17}{1}, \frac{18}{1}, \frac{305}{17}, \frac{323}{18}, \frac{5473}{305}, \frac{5796}{323}, \frac{98209}{5473}, \frac{104005}{5796}, \ldots </math>

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

ApplicationsEdit

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.<ref>Template:Cite journal</ref>