In mathematics, the Lucas sequences and are certain constant-recursive integer sequences that satisfy the recurrence relation where and are fixed integers. Any sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences and More generally, Lucas sequences and represent sequences of polynomials in and with integer coefficients. Famous examples of Lucas sequences include the Fibonacci numbers, Mersenne numbers, Pell numbers, Lucas numbers, Jacobsthal numbers, and a superset of Fermat numbers (see below). Lucas sequences are named after the French mathematician Édouard Lucas. Given two integer parameters and , the Lucas sequences of the first kind and of the second kind are defined by the recurrence relations: and It is not hard to show that for , The above relations can be stated in matrix form as follows: Initial terms of Lucas sequences and are given in the table: The characteristic equation of the recurrence relation for Lucas sequences and is: It has the discriminant and the roots: Thus: Note that the sequence and the sequence also satisfy the recurrence relation. However these might not be integer sequences. When , a and b are distinct and one quickly verifies that It follows that the terms of Lucas sequences can be expressed in terms of a and b as follows The case occurs exactly when for some integer S so that . In this case one easily finds that The ordinary generating functions are When , the Lucas sequences and satisfy certain Pell equations: For any number c, the sequences and with have the same discriminant as and : For any number c, we also have The terms of Lucas sequences satisfy relations that are generalizations of those between Fibonacci numbers and Lucas numbers . For example: Among the consequences is that is a multiple of , i.e., the sequence is a divisibility sequence. This implies, in particular, that can be prime only when n is prime. Another consequence is an analog of exponentiation by squaring that allows fast computation of for large values of n.

About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related courses (1)
CS-101: Advanced information, computation, communication I
Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics a
Related lectures (11)
Generating Functions: Advanced Counting
Covers the definition of generating functions and their application in solving recurrence relations.
Advanced Counting: Generating Functions and Recurrence Relations
Explores generating functions, recurrence relations, and advanced counting techniques.
Counting with Recurrence Relations: Summary
Explores counting with recurrence relations, linear recurrence relations, generating functions, and counting principles.
Show more
Related publications (5)

A Structure Theorem for Level Sets of Multiplicative Functions and Applications

Florian Karl Richter

Given a level set E of an arbitrary multiplicative function f, we establish, by building on the fundamental work of Frantzikinakis and Host [14, 15], a structure theorem that gives a decomposition of 1E1_{E} into an almost periodic and a pseudo-random part ...
2020

DynaProg for Scala

Thierry Coppey

Dynamic programming is an algorithmic technique to solve problems that follow the Bellman’s principle: optimal solutions depends on optimal sub-problem solutions. The core idea behind dynamic programming is to memoize intermediate results into matrices to ...
2013

Algebraic Divisibility Sequences Over Function Fields

Valéry Aurélien Mahé

In this note we study the existence of primes and of primitive divisors in function field analogues of classical divisibility sequences. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields define ...
Australian Mathematical Society2012
Show more
Related concepts (3)
Modular exponentiation
Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie-Hellman Key Exchange and RSA public/private keys. Modular exponentiation is the remainder when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus); that is, c = be mod m. From the definition of division, it follows that 0 ≤ c < m.
Fermat number
In mathematics, a Fermat number, named after Pierre de Fermat, the first known to have studied them, is a positive integer of the form where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... . If 2k + 1 is prime and k > 0, then k itself must be a power of 2, so 2k + 1 is a Fermat number; such primes are called Fermat primes. , the only known Fermat primes are F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537 ; heuristics suggest that there are no more.
Fibonacci sequence
In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn . The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the first few values in the sequence are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.

Graph Chatbot

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.