The Fibonacci Series - Biographies - Édouard Lucas (1842-1891)

Édouard Lucas (1842-1891)

[Édouard Lucas]
Édouard Lucas

François Édouard Anatole Lucas is the 19th century French mathematician for whom the Lucas Series is named. Lucas was born in 1842 in Amiens, France, and was educated at the Ecole Normale in that city. He later worked under Le Verrier at the Paris Observatory. He served as an artillery officer in the Franco-Prussian War (1870-71) and became professor of mathematics at Lycée Saint Louis in Paris after the French defeat. He was later professor of mathematics at the Lycée Charlemagne, also in Paris.

Lucas is most famous for his work with number-theory. He studied the Fibonacci series and the related Lucas series. The Lucas series is defined nearly identically to the Fibonacci series (each number is the sum of the previous two, except for the first two members of the series; f(n) = f(n-2) + f(n-1) ). The difference in the definition is that the Lucas series starts with 2 and 1 rather than 1 and 1. This seems like a small difference at first but once one sees the series continued from 2 and 1, the difference is obvious (see Lots of Lucas Numbers):

Lucas Series:
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, ...

Fibonacci Series:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ...

The Fibonacci and Lucas series are related in many important ways. For one, the ratio of successive Lucas numbers converges to (1.618 ...), just like the Fibonacci series. But before you get too excited about this striking similarity, you should know that this is true of any Fibonacci-like series starting with any pair of numbers; the limit of the growth rate will always be .

Some sources credit Lucas with Binet's Formula, a method for finding Fibonacci numbers. But whether Lucas came up with this formula is open to debate. The mathematician for whom the formula is named, Jacques Philippe Marie Binet, is credited with finding it in 1843, only a year after Édouard Lucas was born. The history of mathematics is full of this type of confusion, so the fact that some credit Lucas with this formula is no surprise.

Édouard Lucas also invented methods of testing for primality (whether or not a number is prime). In 1876 he proved that the Mersenne number 2¹²⁷ - 1 is prime using his own methods. That number still stands as the largest prime ever found without the help a computer, and Lucas's methods for testing for primality are still used today.

The Fibonacci Quarterly, the scholarly publication devoted to the ongoing research into Fibonacci mathematics, frequently publishes new research on Lucas's methods for primes.

[Towers of Hanoi]
Towers of Hanoi

In 1883 Lucas published his famous mathematical game, the Towers of Hanoi, under the pseudonym "M. Claus" ("Claus" is an anagram of "Lucas"). Towers of Hanoi (see right) is a simple puzzle where there are three pegs on a board with discs of ascending size placed from top to bottom around the middle peg. The object is to move all of the discs from one peg to another one at a time in the least number of moves. The only rule is that no disc may be placed on top of a smaller disc at any time.

Random Fibonacci number from the 1st through the 200th:

All contents, unless otherwise specified, are © 1999 by Matt Anderson, Jeffrey Frazier, and Kris Popendorf.
Created by Team 27890 for the 1999 competition.