Lucas number

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The Lucas numbers are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers (both are Lucas sequences). Like the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediate previous terms, i.e. it is a Fibonacci integer sequence. Consequently, the ratio between two consecutive Lucas numbers converges to the golden ratio.

However, the first two Lucas numbers are L0 = 2 and L1 = 1 instead of 0 and 1, and the properties of Lucas numbers are therefore somewhat different from those of Fibonacci numbers.

A Lucas number may thus be defined as follows:

L_n := L(n):= \begin{cases} 2 & \mbox{if } n = 0; \\ 1 & \mbox{if } n = 1; \\ L(n-1)+L(n-2) & \mbox{if } n > 1. \\ \end{cases}

The sequence of Lucas numbers begins:

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ... (sequence A000032 in OEIS)

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[edit] Extension to negative integers

Using Ln-2 = Ln - Ln-1, one can extend the Lucas numbers to negative integers. So we get the following sequence (where values for -5\leq{}n\leq5 are shown): (... -11, 7, -4, 3, -1, 2, 1, 3, 4, 7, 11, ...) . More specifically:

  • L_{-n}=(-1)^nL_n.\!

[edit] Relationship to Fibonacci numbers

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

  • \,L_n = F_{n-1}+F_{n+1}
  • \,L_n^2 = 5 F_n^2 + 4 (-1)^n, and thus as n\, approaches infinity L_n \over F_n\, approaches \sqrt{5}\, .
  • \,F_{2n} = L_n F_n
  • \,F_n = {L_{n-1}+L_{n+1} \over 5}

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.

[edit] Congruence relation

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

[edit] 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, ... (sequence A005479 in OEIS)

Except for the cases n = 0, 4, 8, 16, if Ln is prime then n is prime. The converse is false, however.

[edit] See also

[edit] External links