Fibonacci and Lucas numbers are connected to the "golden" proportion with the help of wonderful mathematical formulas deduced in the 19th century by the French mathematician Binet.
However just as the name of Fibonacci is connected to Fibonacci numbers the name of Binet in modern mathematics associates with the remarkable mathematical formulas connecting Fibonacci and Lucas numbers with the golden proportion.
For deduction of "Binet's formulas" we consider the following elementary formulas for the golden proportion:
Let's remember now that the degrees of the golden proportion are connected by the following identity:
Using the identity (1) we can represent second, third, fourth degrees of the golden proportion in the following "explicit" form:
Whether it is possible to see in these formulas some regularity? First of all we note that each expression for the degree of the golden proportion has the same form:
What are by itself the numerical series À and Â in these formulas? It is easy to be convinced, that the series of A-numbers represents by itself the sequence of numbers 1, 3, 4, 7, 11, ..., and the series of B-numbers represents by itself the sequence of numbers 1, 1, 2, 3, 5, ... . But the former series are Lucas numbers and the latter series are Fibonacci number! From these examples we can guess that in general the formula allowing to present some (n-th) degree of the golden proportion with usage of Lucas number Ln and Fibonacci number Fn should look like the following (and this formula, really, was proved by Binet in the 19th century):
Note that the formula (2) is valid for any integer n.
Using the formula (2) it is possible also very simply to express Lucas number Ln and Fibonacci number Fn through the golden proportion. It is enough for this purpose, using (2), to present the sum or difference of the following degrees of the golden proportion tn + t-n and tn - t-n:
Let's consider now the formulas (3) and (4) for the even values of number of n = 2k. For this purpose we remember one wonderful property of Fibonacci numbers: for the even values of n Fibonacci numbers F2k and F-2k are equal by absolute value but they are opposite by the sign, that is F-2k = - F2k, and the Lucas numbers L2k and L-2k for this case coincide, that is, L-2k = L2k. Then for this case the formulas (3), (4) take the following form:
For the odd values of n = 2k + 1 we have: F-2k-1 = F2k and L-2k-1 = L2k+1. Then for this case the formula (5), (6) are reduced to the following:
The following remarkable expressions for Lucas and Fibonacci follow immediately from the formulas (5) - (8):
The analysis of the formulas (9), (10) gives us a possibility to feel genuine "aesthetic enjoying " and one more to be convinced in a power of man's intellect. Really, we know that the Fibonacci and Lucas numbers always are integers. On the other hand, any degree of the golden proportion is irrational number. It follows from here that the integers Ln and Fn with the help of the formulas (9), (10) express through special irrational numbers.
For example, the Lucas number of 3 (n = 2) according to (9) can be represented as the following:
but the Fibonacci number of 5 (n = 5) as the following:
To be convinced in truth of the expression (11) it is enough to remember that according to (2) the following representations take place:
If now to substitute these expressions in (11) we get that the left-hand part of the identity (11) equals to its right-hand part.
To be convinced in truth of the expression (12) it is enough to remember that according to (2) the following representations take place:
If now to substitute these expressions into (12), we get the following expression for the right-hand part of (12):
it follows from where here a truth of the identity (12).
We can see from this reasoning that Binet's formulas encompass some steep number-theoretic problems located on the joint of integers (Fibonacci and Lucas number) and irrationals (the golden proportion).
Thus, by beginning our "variations" from Fibonacci numbers, we have affected a number of interesting problems relating to algebra and number theory. Certainly, these "variations" do not exhaust all possible applications of Fibonacci numbers. Continuing to look the sections of our Museum our visitor can hereinafter be convinced, how the area of the Fibonacci numbers applications is wide. However it is enough of the looked examples to understand a simple true: just as the simple melody hides in itself incomparably more than it seems at the first listening, the simple mathematical problem about "rabbits reproduction" at the comprehensive consideration allows to look in the wide range of actual mathematical problems.