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File:Lucas number spiral.svg

The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–91), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences.

Definition Edit

Similar to the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediate previous terms, thereby forming a Fibonacci integer sequence. The first two Lucas numbers are L0 = 2 and L1 = 1 as opposed to the first two Fibonacci numbers F0 = 0 and F1 = 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)

The sequence of Lucas numbers is:

$ 2,\;1,\;3,\;4,\;7,\;11,\;18,\;29,\;47,\;76,\;123,\; \ldots\; $OEIS A{{{1}}}.

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 Ln−2 = Ln − Ln−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 numbersEdit

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

  • $ \,L_n = F_{n-1}+F_{n+1}=F_n+2F_{n-1} = F_{n+2}-F_{n-2} $
  • $ \,L_{m+n} = L_{m+1}F_{n}+L_mF_{n-1} $
  • $ \,L_n^2 = 5 F_n^2 + 4 (-1)^n $, and thus as $ n\, $ approaches +∞, the ratio $ \frac{L_n}{F_n} $ approaches $ \sqrt{5}. $
  • $ \,F_{2n} = L_n F_n $
  • $ \,F_{n+k} + (-1)^k F_{n-k} = L_k F_n $
  • $ \,F_n = {L_{n-1}+L_{n+1} \over 5} = {L_{n-3}+L_{n+3} \over 10} $

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. Alternatively, as for $ n>1 $ the magnitude of the term $ (-\varphi)^{-n} $ is less than 1/2, $ 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}}\, , $

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

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

Congruence relationsEdit

If Fn ≥ 5 is a Fibonacci number then no Lucas number is divisible by Fn.

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

Lucas primes Edit

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 A{{{1}}}.

For these ns are

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 A{{{1}}}.

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

Generating seriesEdit

Let

$ \Phi(x) = 2 + x + 3x^2 + 4x^3 + \cdots = \sum_{n = 0}^\infty L_nx^n $

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

$ \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

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


The partial fraction decomposition is given by

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

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

Lucas polynomialsEdit

In the same way as Fibonacci polynomials are derived from the Fibonacci numbers, the Lucas polynomials Ln(x) are a polynomial sequence derived from the Lucas numbers.

See alsoEdit


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