Lucas and Binet's mathematical researches

In the 19th century the interest in Fibonacci numbers and golden section in mathematics increases. The scientific works of the French mathematicians Lucas and Binet are especially noticeable in this respect. In the "Biographic dictionary of the figures in the field of mathematics" (writers Borodino and Bugay) (1979) we can find the brief Lucas biography.

"Francois-Edouard-Anatole Lucas (4.4.1842 - 8.10.1891) is the French mathematician, professor. He was born in └mjen. He worked in the lyceum of Lunle-Gran in Paris. The major works of Lucas fall into number theory and indeterminate analysis. In 1878 Lucas gave the criterion for definition of the primality of Mersenn's numbers of the kind ╠­ = 2 ­ - 1. Applying his method Lucas established, that the number of ╠127 = 2127 - 1 is the prime one. During 75 years this number was the greatest prime number known for science. Also he found the 12th perfect number and formulated a number of interesting mathematical problems. Lucas believed that with the help of machines or other devises the addition is more convenient to perform in the binary number system, than in the decimal one".

Let's give some explanations to Lucas' scientific outcomes. It is well known that the prime numbers are called such numbers, which have not other divisors except for themselves and the unit of 1, namely: 2, 3, 5, 7, 11, 13, ... . Still Pephagoreans proved that a number of the prime numbers is infinite (the proof of this statement is contained in the "Euclidean Elements"). The analysis of the prime numbers and finding out of their distribution in the natural number series is rather difficult problem of number theory. Therefore scientific outcome obtained by Lucas in the field of the prime numbers, doubtlessly, belonged to category of outstanding mathematical achievements.

From the historical point of view it is interesting that Lucas already in the 19th century, that is long before originating modern computers, paid attention on technical advantage of the binary number system, that is, he almost for one century anticipated "John von Neumann Principles" underlying modern electronic computers.

But for our Museum most relevant is the fact that that just Lucas attracted attention to remarkable numeric sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, ..., which was called by him Fibonacci numbers in the honor of the author of this sequence Leonardo Pisano Fibonacci who introduced them in the 13th century. Also Lucas introduced the concept of the generalized Fibonacci numbers, which are computed according to the following recurrent formula:

Gn = Gn-1 + Gn-2,(1)

but for the different initial terms G1 and G2.

For example, the sequence of numbers 3, 8, 11, 19, 30, 49, ... falls to the class of the generalized Fibonacci numbers".

But the main numerical sequence of the type of (1) considered by Lucas is the sequence of numbers 1, 3, 4, 7, 11, 18, 29, 47, ... given with the following recurrent formula:

Ln = Ln-1 + Ln-2,(2)

for the initial terms L1 = 1 and L2 = 3.

In the honor of Lucas this numerical sequence was called "Lucas numbers". Note that Lucas numbers have the same significance for mathematics, as well as the classical Fibonacci numbers.

After Lucas the mathematical works on Fibonacci numbers, according to saying of one mathematician, "begun to propagate as Fibonacci's rabbits " - and in this the historical Lucas merit for Fibonacci number theory consists!

About the other 19th century enthusiast of Fibonacci numbers, the French mathematician Binet (Jacques Philippe Marie Binet), we have the following information. He was born on February 2, 1776 in Renje and died on May 12, 1856 in Paris.

Jacques Philippe Marie Binet (1776-1856)
Jacques Philippe Marie Binet (1776-1856)

Binet graduated from the Polytechnic School in Paris and after its graduation in 1806 he worked at the Bridges and Roads Department of the French government.

He became as a teacher of the Polytechnic school in 1807 and in one year became as an assistant-professor of the applied analysis and descriptive geometry.

Binet investigated foundations of matrix theory and his works in this direction were continued then by other researchers. He discovered in 1812 the rule of matrix multiplication and already this discovery glorified his name more, than other his works.

Except for mathematics Binet worked and in other areas. He published many articles on mechanics, mathematics and astronomy. In mathematics Binet introduced the notion of the "beta function"; also he considered the linear difference equations with alternating coefficients and established some metric properties of conjugate diameters and so on.

Among different honors obtained by Binet even at his life it is necessary to mention that he was selected to the Parisian Academy of sciences in 1843.

Most interesting for our Museum is the fact that Binet studied the linear difference equations; the Fibonacci recurrent equation is their particular case. Apparently, just this fascination resulted him in the famous Binet's formulas, which connect Fibonacci and Lucas numbers with the golden proportion. Let's remind, that Binet's formulas in mathematics are perceived as the following group of the formulas:

(3)
(4)

where Ln and Fn are Lucas and Fibonacci numbers respectively, is the golden proportion.

What mean the formulas (3), (4)? The formula (3) means, that the n-th Lucas number Ln can be presented or as the sum of the golden proportion degrees tn + t -n for the even values of n = 2k or as the difference tn - t -n if n = 2k + 1. The formula (4) asserts that for representation of the n-th Fibonacci number Fn it is necessary to make the same, that is, to compute the sum tn + t -n for the odd values of n = 2k + 1 or the difference tn - t -n if n = 2k and then to divide them by the irrational number .

Lucas and Binet's investigations became by that launch pad for the group of the American 20th century mathematicians who organized in 1963 the Fibonacci Association and begun to issue "The Fibonacci Quarterly" since 1963. We will tell about the Fibonacci Association on the next pages of our Museum. Follow us!