# François Édouard Anatole Lucas

### Quick Info

Amiens, France

Paris, France

**Edouard Lucas**was a French mathematician who is best known for his results in number theory. He studied the Fibonacci sequence and devised the test for Mersenne primes still used today.

### Biography

**Édouard Lucas**was educated at the École Normale in Amiens. After this he worked at the Paris Observatory under Le Verrier.

During the Franco-Prussian War (1870-1871) Lucas served as an artillery officer. After the French were defeated, Lucas became professor of mathematics at the Lycée Saint Louis in Paris. He later became professor of mathematics at the Lycée Charlemagne, also in Paris.

Lucas is best known for his results in number theory: in particular he studied the Fibonacci sequence and the associated Lucas sequence is named after him. He gave the well-known formula for the Fibonacci numbers

$\sqrt 5 f_{n} = \left(\large\frac{1 - \sqrt 5}{2}\right)^{n}\normalsize - \left(\large\frac{1 - \sqrt 5}{2}\right)^{n}$.

Lucas also devised methods of testing primality, essentially those used today. In 1876 he used his methods to prove that the Mersenne number $2^{127} - 1$ is prime. This remains the largest prime number discovered without the aid of a computer.
The Lucas test for primes was refined by Lehmer in 1930. It works as follows. Define the sequence

$S_{2} = 4, S_{3} = 14, S_{4} = 194, . . .$

where for $n > 2, S_{n}$ is defined inductively by
$S_{n} = S_{n-1}^{2} - 2$.

The Lucas-Lehmer test states that a Mersenne number $M_{p} = 2^{p} - 1$, with $p > 2$, is prime if and only if $M_{p}$ divides $S_{p}$.
Lucas showed that $S_{127}$ is divisible by $M_{127}$ thus showing that $M_{127}$ is prime. This was a extremely difficult calculation since $M_{127}$ is a big number and $S_{127}$ is unbelievably large. In fact

$M_{127}$ = 170141183460469231731687303715884105727

and Lucas was only able to perform the calculation since he showed that $S_{127}$ is divisible by $M_{127}$ without calculating $S_{127}$.
Lucas is also well known for his invention of the Tower of Hanoi puzzle and other mathematical recreations. The Tower of Hanoi puzzle appeared in 1883 under the name of M. Claus. Notice that Claus is an anagram of Lucas! His four volume work on recreational mathematics

*Récréations mathématiques*Ⓣ (1882-94) has become a classic.

Lucas died as the result of a freak accident at a banquet when a plate was dropped and a piece flew up and cut his cheek. He died of erysipelas a few days later.

### References (show)

- N T Gridgeman, Biography in
*Dictionary of Scientific Biography*(New York 1970-1990). See THIS LINK. - D Harkin, On the Mathematical Works of François Edouard Anatole Lucas,
*Enseignement mathématique***3**(1957), 276-288. - M R Williams, Edouard Lucas vindicated,
*Ann. Hist. Comput.***6**(4) (1984), 404. https://www.google.com/search?tbm=bks&q=Luca+R%C3%A9cr%C3%A9ations+math%C3%A9matiques

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Written by J J O'Connor and E F Robertson

Last Update December 1996

Last Update December 1996