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. Much like the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediate predecessors. Consequently, the ratio between two consecutive Lucas numbers converges to the golden ratio. In mathematics, an integer sequence is a sequence (i. ... François Édouard Anatole Lucas (April 4, 1842 in Amiens - October 3, 1891) was a French mathematician. ... In mathematics, the Fibonacci numbers form a sequence defined recursively by: That is, after two starting values,each number is the sum of the two preceding numbers. ... The golden section is a line segment sectioned into two according 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 the Lucas sequence are therefore somewhat different from those of the Fibonacci sequence.

A Lucas number may thus be defined as follows:

The sequence of Lucas numbers begins:

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

2 (two) is a number, numeral, and glyph. ... Look up one in Wiktionary, the free dictionary. ... This article discusses the number three. ... This article discusses the number Four. ... // Seven is the fourth prime number. ... 11 (eleven) is the natural number following 10 and preceding 12. ... 18 (eighteen) is the natural number following 17 and preceding 19. ... 29 (twenty-nine) is the natural number following 28 and preceding 30. ... 47 (forty-seven) is the natural number following 46 and followed by 48. ... 76 is the natural number following 75 and preceding 77. ... 123 is the natural number following 122 and preceding 124. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

Extension to negative integers

Using L_n-2 = L_n - L_n-1, one can extend the Lucas numbers to negative integers. So we get: ... -11, 7, -4, 3, -1, 2, 1, 3, 4, 7, 11, ... and L_-n = (-1)ⁿL_n.

Relationship to Fibonacci numbers

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

Their closed formula is given as:

where is the Golden ratio.

Lucas primes

A Lucas prime is a Lucas number that is prime. The first few Lucas primes are In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors. ...

2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, ... (sequence A005479 in OEIS) The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

Except for the cases n = 0, 4, 8, 16, if L_n is prime then n is prime. The converse is false, however. In traditional logic Conversion is a form of immediate inference in which from a given categorical proposition another proposition is inferred which has as its subject the predicate of the original proposition, and has as its predicate the subject of the original proposition, with the quality of the proposition remaining...

See also Fibonacci prime. A Fibonacci prime is a Fibonacci number that is prime. ...

External links

• MathWorld
• Dr Ron Knott
• Lucas numbers and the Golden Section