Not to be confused with Lucas sequences, the general class of sequences to which the Lucas numbers belong.

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.[1] 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.[2] 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.[3]

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, ... .

which coincides for example with the number of independent vertex sets for cyclic graphs C_n of length n\geq2.[1]

Definition

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.

Extension to negative integers

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\leq{}n\leq5 are shown).

The formula for terms with negative indices in this sequence is

L_{-n}=(-1)^nL_n.\!

Relationship to Fibonacci numbers

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} = F_{n+2}-F_{n-2}.
  • L_{n+k} = L_{n+1}F_k + L_n F_{k-1}.
  • F_{n+k} = L_k F_n - (-1)^k F_{n-k}; in particular when k=n, F_{2n} = L_n F_n.
  • 2F_{n+k} = L_n F_k + L_k F_n.
  • L_{2n} = 5 F_n^2 + 2(-1)^n = L_n^2 - 2(-1)^n, so \lim_{n\to\infty} \frac{L_n}{F_n}=\sqrt{5}.
  • L_n - \sqrt{5} F_n = \frac{2}{(-\varphi)^n} \to 0.
  • L_{n+k} - (-1)^k L_{n-k} = 5 F_n F_k; in particular when k=1, 5F_n = L_{n+1}+L_{n-1} = 2L_{n+1} - L_n = L_n + 2L_{n-1}, while when k=n, the identity above for L_{2n} is obtained.

Their closed formula is given as:

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 ,

where \varphi is the golden ratio. For n>1 the magnitude of the term (1-\varphi)^n is less than 1/2, therefore L_n is the closest integer to \varphi^n or, equivalently, the integer part of \varphi^n+1/2, also written as \lfloor \varphi^n+1/2 \rfloor.

Combining the above with Binet's formula,

F_n = \frac{\varphi^n - (1-\varphi)^n}{\sqrt 5} = \frac{\varphi^n - (-\varphi)^{-n}}{\sqrt 5} \, ,

a formula for \varphi^n is obtained:

\varphi^n = {L_n + F_n \sqrt 5 \over 2}\, .

For integers n ≥ 2, we also get:

\varphi^n = L_n - (1-\varphi)^n = L_n - (-1)^n L_n^{-1} - L_n^{-3} + R

with remainder R satisfying

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

Lucas identities

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

\sum_{k=0}^n L_k = L_{n+2} - 1
\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 \textstyle F_n=\frac{L_{n-1}+L_{n+1}}{5}.

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

where L'_n=L_n except for L'_0=1.

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

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

Generating function

The ordinary generating function of the sequence of Lucas numbers is the power series

\Phi(x) = \sum_{k=0}^\infty L_k x^k = 2 + x + 3x^2 + 4x^3 + 7x^4 + 11x^5 + \cdots.

This series is convergent for any complex number x satisfying |x| < 1/\varphi \approx 0.618, and its sum has a simple closed form:

\Phi(x)=\frac{2-x}{1-x-x^2}.

This can be proved by multiplying by (1-x-x^2):

\begin{align}
(1 - x- x^2) \Phi(x)
  &= \sum_{k=0}^{\infty} L_k x^k - \sum_{k=0}^{\infty} L_k x^{k+1} - \sum_{k=0}^{\infty} L_k x^{k+2} \\
  &= \sum_{k=0}^{\infty} L_k x^k - \sum_{k=1}^{\infty} L_{k-1} x^k - \sum_{k=2}^{\infty} L_{k-2} x^k \\
  &= 2x^0 + 1x^1 - 2x^1 + \sum_{k=2}^{\infty} (L_k - L_{k-1} - L_{k-2}) x^k \\
  &= 2 - x,
\end{align}

where all terms involving x^k for k \ge 2 cancel out because of the defining Lucas numbers recurrence relation.

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

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

\Phi(x) satisfies the functional equation

\Phi(x) - \Phi\!\left(-\frac{1}{x}\right) = 2

As the generating function for the Fibonacci numbers is given by

s(x) = \frac{x}{1 - x - x^2}

we have

s(x) + \Phi(x) = \frac{2}{1 - x - x^2}

which proves that

F_n + L_n = 2F_{n+1},

and

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

proves that

5F_n + L_n = 2L_{n+1}

The partial fraction decomposition is given by

\Phi(x) = \frac{1}{1 - \phi x} + \frac{1}{1 - \psi x}

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

This can be used to prove the generating function, as

\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)

Using x equal to any of 0.01, 0.001, 0.0001, etc. lays out the first Lucas numbers in the decimal expansion of \Phi(x). For example, \Phi(0.001) = \frac{1.999}{0.998999} = \frac{1999000}{998999} = 2.001003004007011018029047\ldots.

Congruence relations

If F_n\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.

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, 3010349, 54018521, 370248451, 6643838879, ... .

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, ... .

As of September 2015, the largest confirmed Lucas prime is L148091, which has 30950 decimal digits.[4] As of August 2022, the largest known Lucas probable prime is L5466311, with 1,142,392 decimal digits.[5]

If Ln is prime then n is 0, prime, or a power of 2.[6] L2m is prime for m = 1, 2, 3, and 4 and no other known values of m.

Lucas polynomials

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.

Continued fractions for powers of the golden ratio

For all but the smallest values of n, the integer Ln very closely approximates the n-th power of the golden ratio, \varphi^n. Furthermore, close rational approximations for powers of the golden ratio can be obtained from their continued fractions.

For positive integers n, the continued fractions are:

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

For example:

\varphi^5 = [11; 11, 11, 11, \ldots]

is the limit of

\frac{11}{1}, \frac{122}{11}, \frac{1353}{122}, \frac{15005}{1353}, \ldots

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

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

is the limit of

\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

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

Applications

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.[7]

See also

References

  1. ^ Weisstein, Eric W. "Lucas Number". mathworld.wolfram.com. Retrieved 2020-08-11.
  2. ^ Parker, Matt (2014). "13". Things to Make and Do in the Fourth Dimension. Farrar, Straus and Giroux. p. 284. ISBN 978-0-374-53563-6.
  3. ^ Parker, Matt (2014). "13". Things to Make and Do in the Fourth Dimension. Farrar, Straus and Giroux. p. 282. ISBN 978-0-374-53563-6.
  4. ^ "The Top Twenty: Lucas Number". primes.utm.edu. Retrieved 6 January 2022.
  5. ^ "Henri & Renaud Lifchitz's PRP Top - Search by form". www.primenumbers.net. Retrieved 6 January 2022.
  6. ^ Chris Caldwell, "The Prime Glossary: Lucas prime" from The Prime Pages.
  7. ^ Swinton, Jonathan; Ochu, Erinma; null, null (2016). "Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment". Royal Society Open Science. 3 (5). Bibcode:2016RSOS....360091S. doi:10.1098/rsos.160091. PMC 4892450. PMID 27293788