Died: 3 October 1891 in Paris, France

Edouard 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

√5 f_n = ((1 + √5)/2)ⁿ - ((1 - √5)/2)ⁿ.

Lucas also devised methods of testing primality, essentially those used today. In 1876 he used his methods to prove that the Mersenne number 2¹²⁷ - 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

S₂ = 4, S₃ = 14, S₄ = 194, . . .

where for n >2, S_n is defined inductively by

S_n = S_n-1² - 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 S₁₂₇ is divisible by M₁₂₇ thus showing that M₁₂₇ is prime. This was a extremely difficult calculation since M₁₂₇ is a big number and S₁₂₇ is unbelievably large. In fact

M₁₂₇ = 170141183460469231731687303715884105727

and Lucas was only able to perform the calculation since he showed that S₁₂₇ is divisible by M₁₂₇ without calculating S₁₂₇.

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.

1. N T Gridgeman, Biography in Dictionary of Scientific Biography (New York 1970-1990). 
   http://www.encyclopedia.com/doc/1G2-2830902693.html

Articles:

1. D Harkin, On the Mathematical Works of François Edouard Anatole Lucas, Enseignement mathématique 3 (1957), 276-288.
2. M R Williams, Edouard Lucas vindicated, Ann. Hist. Comput. 6 (4) (1984), 404.

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