The Lucas numbers or Lucas series are an
integer sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers.
An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For ...
named after the mathematician
François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related
Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
s. Lucas numbers and Fibonacci numbers form complementary instances of
Lucas sequence
In mathematics, the Lucas sequences U_n(P,Q) and V_n(P, Q) are certain constant-recursive integer sequences that satisfy the recurrence relation
: x_n = P \cdot x_ - Q \cdot x_
where P and Q are fixed integers. Any sequence satisfying this recu ...
s.
The Lucas series 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. This produces a sequence where the ratios of successive terms approach the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
, and in fact the terms themselves are
rounding
Rounding means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation. For example, replacing $ with $, the fraction 312/937 with 1/3, or the expression with .
Rounding is often done to obta ...
s of integer powers of the golden ratio. 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.
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 ....
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
and
, which differs from the first two Fibonacci numbers
and
. Though closely related in definition, Lucas and Fibonacci numbers exhibit distinct properties.
The Lucas numbers may thus be defined as follows:
:
(where ''n'' belongs to the natural numbers)
The sequence of the first twelve Lucas numbers is:
:
.
All Fibonacci-like integer sequences appear in shifted form as a row of the
Wythoff array
In mathematics, the Wythoff array is an infinite matrix of integers derived from the Fibonacci sequence and named after Dutch mathematician Willem Abraham Wythoff. Every positive integer occurs exactly once in the array, and every integer sequence ...
; 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
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
.
Extension to negative integers
Using
, 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 many identities. Among these are the following:
*
*
*
*
*
*
, so
.
*
*
; in particular,
, so
.
Their
closed formula
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th ro ...
is given as:
:
where
is the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
. 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
.
Combining the above with
Binet's formula
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
,
:
a formula for
is obtained:
:
For integers ''n'' ≥ +2, we also get:
:
with remainder ''R'' satisfying
:
.
Lucas identities
Many of the Fibonacci identities have parallels in Lucas numbers. For example, the
Cassini identity becomes
:
Also
:
:
:
where
.
:
where
except for
.
For example if ''n'' is odd,
and
Checking,
, and
Generating function
Let
:
be the
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
of the Lucas numbers. By a direct computation,
:
which can be rearranged as
:
gives the generating function for the
negative indexed Lucas numbers,
, and
:
satisfies the
functional equation
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
:
As the
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
for the Fibonacci numbers is given by
:
we have
:
which proves that
:
And
:
proves that
:
The
partial fraction decomposition
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as ...
is given by
:
where
is the golden ratio and
is its conjugate.
This can be used to prove the generating function, as
:
Congruence relations
If
is a Fibonacci number then no Lucas number is divisible by
.
is congruent to 1
modulo
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is t ...
if
is prime, but some composite values of
also have this property. These are the
Fibonacci pseudoprimes.
is congruent to 0 modulo 5.
Lucas primes
A Lucas prime is a Lucas number that is
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. 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, ''L''
4 = 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, ... .
, the largest confirmed Lucas prime is ''L''
148091, which has 30950 decimal digits. , the largest known Lucas
probable prime
In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different specific con ...
is ''L''
5466311, with 1,142,392 decimal digits.
If ''L
n'' is prime then ''n'' is 0, prime, or a power of 2. ''L''
2''m'' is prime for ''m'' = 1, 2, 3, and 4 and no other known values of ''m''.
Lucas polynomials
In the same way as
Fibonacci polynomial In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials.
Definition
Th ...
s are derived from the
Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
s, the
Lucas polynomials In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials.
Definition
The ...
are a
polynomial 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. Polynomial sequences are a topic of interest in en ...
derived from the Lucas numbers.
Continued fractions for powers of the golden ratio
Close rational approximations for powers of the golden ratio can be obtained from their
continued fraction
In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...
s.
For positive integers ''n'', the continued fractions are:
:
:
.
For example:
:
is the limit of
:
with the error in each term being about 1% of the error in the previous term; and
:
is the limit of
:
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.
See also
*
Generalizations of Fibonacci numbers In mathematics, the Fibonacci numbers form a sequence defined recursively by:
:F_n =
\begin
0 & n = 0 \\
1 & n = 1 \\
F_ + F_ & n > 1
\end
That is, after two starting values, each number is the sum of the two preceding numbers.
The Fibonacci sequ ...
References
External links
*
*
*
*
The Lucas Numbers, Dr Ron Knott
A Lucas Number Calculator can be found here.*
{{series (mathematics)
Integer sequences
Fibonacci numbers
Recurrence relations
Unsolved problems in mathematics
bn:লà§à¦•à¦¾à¦¸ ধারা
fr:Suite de Lucas
he:סדרת לוק×ס
pt:Sequência de Lucas