Lucas number
Encyclopedia
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 number
s. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequence
s.
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:
The sequence of Lucas numbers begins:
The formula for terms with negative indices in this sequence is
Their closed formula
is given as:
where
is the Golden ratio
. Alternatively, as for
the magnitude of the term 
is less than 1/2, 
is the closest integer to 
or, equivalently, the integer part of 
, also written as 
.
Conversely,
.
. The first few Lucas primes are
If Ln is prime then n is either 0, prime, or a power of 2. L
is prime for 
= 1, 2, 3, and 4 and no other known values of 
.
derived from the Lucas numbers in the same way as Fibonacci polynomials are derived from the Fibonacci number
s. Lucas polynomials are defined by the following recurrence relation:
Lucas polynomials can be expressed in terms of Lucas sequence
s as
The first few Lucas polynomials are:
The Lucas numbers are recovered by evaluating the polynomials at x = 1. The degree of Ln(x) is n. The ordinary generating function for the sequence is
Integer sequence
In mathematics, an integer sequence is a sequence of integers.An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms...
named after the mathematician François Édouard Anatole Lucas
Edouard Lucas
François Édouard Anatole Lucas was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequences and Lucas numbers are named after him.-Biography:...
(1842–1891), who studied both that sequence and the closely related Fibonacci number
Fibonacci number
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....
s. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequence
Lucas sequence
In mathematics, the Lucas sequences Un and Vn are certain integer sequences that satisfy the recurrence relationwhere P and Q are fixed integers...
s.
Definition
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 convergesLimit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
to the golden ratio
Golden ratio
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
. 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:
The sequence of Lucas numbers begins:
- 2, 1, 3, 4, 7, 1111 (number)11 is the natural number following 10 and preceding 12.Eleven is the first number which cannot be counted with a human's eight fingers and two thumbs additively. In English, it is the smallest positive integer requiring three syllables and the largest prime number with a single-morpheme name...
, 1818 (number)18 is the natural number following 17 and preceding 19.In speech, the numbers 18 and 80 are sometimes confused. When carefully enunciated, they differ in which syllable is stressed: 18 vs 80 . However, in dates such as 1864, or when contrasting numbers in the teens, such as 17, 18, 19, the stress...
, 2929 (number)29 is the natural number following 28 and preceding 30.-In mathematics:It is the tenth prime number, and also the fourth primorial prime. It forms a twin prime pair with thirty-one, which is also a primorial prime. Twenty-nine is also the sixth Sophie Germain prime. It is also the sum of three...
, 4747 (number)47 is the natural number following 46 and preceding 48.-In mathematics:Forty-seven is the fifteenth prime number, a safe prime, the thirteenth supersingular prime, and the sixth Lucas prime. Forty-seven is a highly cototient number...
, 7676 (number)76 is the natural number following 75 and preceding 77.-In mathematics:Seventy-six is a Lucas number, an automorphic number, a nontotient, a 14-gonal number, and a centered pentagonal number....
, 123123 (number)123 is the natural number following 122 and preceding 124.-In mathematics:* 123 is a Lucas number...
, ... .
Extension to negative integers
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
for 
are shown).
The formula for terms with negative indices in this sequence is
Relationship to Fibonacci numbers
The Lucas numbers are related to the Fibonacci numbers by the identities-
-
, and thus as 
approaches +∞Extended real number lineIn mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ . The projective extended real number system adds a single object, ∞ and makes no distinction between "positive" or "negative" infinity...
, the ratio
approaches 
-
-
-
Their closed formula
Closed-form expression
In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a bounded number of certain "well-known" functions...
is given as:
where
is the Golden ratioGolden ratio
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
. Alternatively, as for
the magnitude of the term is less than 1/2, is the closest integer to or, equivalently, the integer part of , also written as .Conversely,
.Congruence relation
Ln is congruent to 1 mod n if n is prime, but some composite values of n also have this property.Lucas primes
A Lucas prime is a Lucas number that is primePrime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
. The first few Lucas primes are
- 2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, ... .
If Ln is prime then n is either 0, prime, or a power of 2. L
is prime for = 1, 2, 3, and 4 and no other known values of .Lucas polynomials
The Lucas polynomials Ln(x) are a polynomial sequencePolynomial sequence
In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial...
derived from the Lucas numbers in the same way as Fibonacci polynomials are derived from the Fibonacci number
Fibonacci number
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....
s. Lucas polynomials are defined by the following recurrence relation:
Lucas polynomials can be expressed in terms of Lucas sequence
Lucas sequence
In mathematics, the Lucas sequences Un and Vn are certain integer sequences that satisfy the recurrence relationwhere P and Q are fixed integers...
s as
The first few Lucas polynomials are:
The Lucas numbers are recovered by evaluating the polynomials at x = 1. The degree of Ln(x) is n. The ordinary generating function for the sequence is
External links
- Dr Ron Knott
- Lucas numbers and the Golden Section
- A Lucas Number Calculator can be found here.
- A Tutorial on Generalized Lucas Numbers
- Lucas Numbers from The On-Line Encyclopedia of Integer Sequences.