Edouard Lucas
Encyclopedia
François Édouard Anatole Lucas (4 April 1842 – 3 October 1891) was a French
mathematician
. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequence
s and Lucas number
s are named after him.
. He worked in the Paris observatory and later became a professor of mathematics in Paris. In the meantime he served in the army.
In 1875, Lucas posed a challenge to prove that the only solution of the Diophantine equation
:
with N > 1 is when N = 24 and M = 70. This is known as the cannonball problem, 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 hyperelliptic functions) was found for this remarkable fact, which has relevance to the bosonic string theory
in 26 dimensions. Elementary proof
s have been more recently published.
He devised methods for testing the primality
of numbers. In 1857, at age 15, Lucas began testing the primality of 2127 − 1 by hand, using Lucas Sequences. In 1876, after 19 years of testing, he finally proved that 2127 − 1 was prime; this would remain the largest known Mersenne prime
for three-quarters of a century. This may stand forever as the largest prime number proven by hand. Later Derrick Henry Lehmer
refined Lucas' primality tests and obtained the Lucas-Lehmer test.
He worked on the development of the umbral calculus
.
Lucas was also interested in recreational mathematics
. He found an elegant binary
solution to the Baguenaudier
puzzle. He more notably invented the Tower of Hanoi
puzzle, 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 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 septicemia. Lucas was only 49 years old when he died.
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. Lucas is known for his study of the Fibonacci sequence. The related 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 and Lucas number
Lucas number
The Lucas numbers are an integer sequence named after the mathematician François Édouard Anatole Lucas , who studied both that sequence and the closely related Fibonacci numbers...
s are named after him.
Biography
Lucas was educated at the École Normale SupérieureÉcole Normale Supérieure
The École normale supérieure is one of the most prestigious French grandes écoles...
. He worked in the Paris observatory and later became a professor of mathematics in Paris. In the meantime he served in the army.
In 1875, Lucas posed a challenge to prove that the only solution of the Diophantine equation
Diophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
:
with N > 1 is when N = 24 and M = 70. This is known as the cannonball problem, since it can be visualized as the problem of taking a square arrangement of cannonballs on the ground and building a square pyramid
Square pyramid
In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry.- Johnson solid :...
out of them. It was not until 1918 that a proof (using hyperelliptic functions) was found for this remarkable fact, which has relevance to the bosonic string theory
Bosonic string theory
Bosonic string theory is the original version of string theory, developed in the late 1960s.In the early 1970s, supersymmetry was discovered in the context of string theory, and a new version of string theory called superstring theory became the real focus...
in 26 dimensions. Elementary proof
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. For some time it was thought that certain theorems, like the prime number theorem, could only be...
s have been more recently published.
He devised methods for testing the primality
Prime 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...
of numbers. In 1857, at age 15, Lucas began testing the primality of 2127 − 1 by hand, using Lucas Sequences. In 1876, after 19 years of testing, he finally proved that 2127 − 1 was prime; this would remain the largest known Mersenne prime
Mersenne prime
In mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...
for three-quarters of a century. This may stand forever as the largest prime number proven by hand. Later Derrick Henry Lehmer
Derrick Henry Lehmer
Derrick Henry "Dick" Lehmer was an American mathematician who refined Édouard Lucas' work in the 1930s and devised the Lucas–Lehmer test for Mersenne primes...
refined Lucas' primality tests and obtained the Lucas-Lehmer test.
He worked on the development of the umbral calculus
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 and are sometimes called Blissard's symbolic method...
.
Lucas was also interested in recreational mathematics
Recreational mathematics
Recreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games.Not all problems in this field require a knowledge of advanced mathematics, and thus, recreational mathematics often attracts the curiosity of non-mathematicians, and inspires their further study...
. He found an elegant binary
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
solution to the Baguenaudier
Baguenaudier
right|thumb|200pxBaguenaudier is a mechanical puzzle featuring a double loop of string which must be disentangled from a sequence of rings on interlinked pillars. The puzzle is thought to have been invented originally in China...
puzzle. He more notably invented the Tower of Hanoi
Tower of Hanoi
The Tower of Hanoi or Towers of Hanoi, also called the Tower of Brahma or Towers of Brahma, is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod...
puzzle, which he marketed under the nickname N. Claus de Siam, an anagram
Anagram
An anagram is a type of word play, the result of rearranging the letters of a word or phrase to produce a new word or phrase, using all the original letters exactly once; e.g., orchestra = carthorse, A decimal point = I'm a dot in place, Tom Marvolo Riddle = I am Lord Voldemort. Someone who...
of Lucas d'Amiens, and published for the first time a description of the Dots and Boxes
Dots and Boxes
Dots and Boxes is a pencil and paper game for two players first published in 1889 by Édouard Lucas.Starting with an empty grid of dots, players take turns, adding a single...
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 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 septicemia. Lucas was only 49 years old when he died.
Works
- Recherches Sur Plusieurs Ouvrages De Léonard De Pise Et Sur Diverses Questions D’Arithmétique Supérieure (1877)
- Théorie des nombres, Tome Premier (1891)
- Récréations mathématiques (1894)
- L'arithmétique amusante (1895)