__NOTOC__ François Édouard Anatole Lucas (; 4 April 1842 – 3 October 1891) was a French

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

. Lucas is known for his study of the

Fibonacci sequence 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 ...

. The related

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 rec ...

s and

Lucas number The Lucas numbers or Lucas series 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. Lucas numbers and Fibonacci n ...

s are named after him.

Biography

Lucas was born in

Amiens Amiens (English: or ; ; pcd, Anmien, or ) is a city and commune in northern France, located north of Paris and south-west of Lille. It is the capital of the Somme department in the region of Hauts-de-France. In 2021, the population of ...

and educated at the

École Normale Supérieure École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, S ...

. He worked in the Paris Observatory and later became a professor of mathematics at the Lycée Saint Louis and the Lycée Charlemagne in

Paris Paris () is the Capital city, capital and List of communes in France with over 20,000 inhabitants, most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), ma ...

. Lucas served as an artillery officer in the French Army during the Franco-Prussian War of 1870–1871. In 1875, Lucas posed a challenge to prove that the only solution of the Diophantine equation: :

\sum_^ n^2 = M^2\;

with ''N'' > 1 is when ''N'' = 24 and ''M'' = 70. This is known as the

cannonball problem In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be a ...

, since it can be visualized as the problem of taking a square arrangement of cannonballs on the ground and building a square pyramid out of them. It was not until 1918 that a proof (using elliptic functions) was found for this remarkable fact, which has relevance to the

bosonic string theory Bosonic string theory is the original version of string theory, developed in the late 1960s and named after Satyendra Nath Bose. It is so called because it contains only bosons in the spectrum. In the 1980s, supersymmetry was discovered in the co ...

in 26 dimensions. More recently,

elementary proof In mathematics, an elementary proof is a mathematical proof that only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis. Historically, it was once thought that certain ...

s have been published. He devised methods for testing the

primality 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 ...

of numbers. In 1857, at age 15, Lucas began testing the primality of 2¹²⁷ − 1 by hand, using

s. In 1876, after 19 years of testing, he finally proved that 2¹²⁷ − 1 was prime; this would remain the largest known

Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...

for three-quarters of a century. This may stand forever as the largest prime number proven by hand. Later

Derrick Henry Lehmer Derrick Henry "Dick" Lehmer (February 23, 1905 – May 22, 1991), almost always cited as D.H. Lehmer, was an American mathematician significant to the development of computational number theory. Lehmer refined Édouard Lucas' work in the 1930s and ...

refined Lucas's

primality tests A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating wh ...

and obtained the Lucas–Lehmer primality test. He worked on the development of the

umbral calculus In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain "shadowy" techniques used to "prove" them. These techniques were introduced by John Blis ...

. Lucas was also interested in

recreational mathematics Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...

. He found an elegant

binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that ta ...

solution to the

Baguenaudier Baguenaudier (; French for "time-waster"), also known as the Chinese rings, Cardan's suspension, Cardano's rings, Devil's needle or five pillars puzzle, is a disentanglement puzzle featuring a loop which must be disentangled from a sequence of ...

puzzle. He also invented the

Tower of Hanoi The Tower of Hanoi (also called The problem of Benares Temple or Tower of Brahma or Lucas' Tower and sometimes pluralized as Towers, or simply pyramid puzzle) is a mathematical game or puzzle consisting of three rods and a number of disks of v ...

puzzle in 1883, which he marketed under the nickname ''N. Claus de Siam'', an anagram of ''Lucas d'Amiens'', and published for the first time a description of the Dots and Boxes game in 1889. Lucas died in unusual circumstances. At the banquet of the annual congress of the ''Association française pour l'avancement des sciences'', a waiter dropped some

crockery Tableware is any dish or dishware used for setting a table, serving food, and dining. It includes cutlery, glassware, serving dishes, and other items for practical as well as decorative purposes. The quality, nature, variety and number of obj ...

and a piece of broken plate cut Lucas on the cheek. He died a few days later of a severe skin inflammation probably caused by

sepsis Sepsis, formerly known as septicemia (septicaemia in British English) or blood poisoning, is a life-threatening condition that arises when the body's response to infection causes injury to its own tissues and organs. This initial stage is follo ...

. He was only 49 years old.

Works

* ''Recherches Sur Plusieurs Ouvrages De Léonard De Pise Et Sur Diverses Questions D’Arithmétique Supérieure'' (1877)
''Récréations scientifiques''
(1880)
''Théorie des nombres''
Tome Premier (1891)
''Récréations mathématiques''
(1894)
''L'arithmétique amusante''
(1895)

References

* *. *Harkin, D. “On the Mathematical Works of Francois-Édouard-Anatole Lucas, ''Enseignement mathematique'', 2nd ser., 3 (1957), 276–288.

External links

*
Scans of Lucas's original Tower of Hanoi puzzle in French, with translations
by Clark Kimberling
Édouard Lucas
{{DEFAULTSORT:Lucas, Edouard 1842 births 1891 deaths 19th-century French mathematicians Number theorists École Normale Supérieure alumni Fibonacci numbers Recreational mathematicians Mathematics popularizers Deaths from sepsis People from Amiens

Biography

Works

See also

References

External links