Golden Proportion equation

Since the oldest times the solution of algebraic equations attracted a special attention of mathematicians and this relevant mathematical problem promoted to development of algebra. The rules of the solution of algebraic equations of the 1-st and 2-nd degree were known in a remote antiquity. But the formulas for the radicals of algebraic equations of the 3-rd and 4-th degree were retrieved only in the 16-17th centuries. For the radicals of the equations of the 5-th and more degrees the general formula does not exist. The solution of the general algebraic equations in radicals had resulted of the French 19th century mathematician E. Galois to a general theory of algebraic equations called theory of Galois.

Let's consider once again algebraic equation of the "golden" proportion:

 (1)

It is the algebraic equation of the 2-nd degree and each schoolboy can solve it, that is, find its roots. But there is a question: whether there are algebraic equations of more high degrees having the "golden" proportion as their root? For the answer this question we will carry out the following reasoning, by taking the simplest equation of the "golden" proportion assigned (1) as the initial one.

Let's multiply both parts of the equation (1) by x, then we have:

 (2)

It follows from (1) that the value x can be presented as the following: Let's substitute now this value for the variable x to the equation (2); then we will get the following equation of the 3-rd order:

 (3)

On the other hand, if we substitute to the equation (2) the expression for x2, assigned (1), we will get one more equation of the 3-rd order:

 (4)

Thus, we have got two new equations of the 3-rd degree, which radicals are the golden proportion.

Let's represent now the equation of the golden proportion (1) in the following form:

 (5)

Then we can take the expression (1) for x2, and also to take the expressions (3) or (4) for x3. Substituting them to the expression (5), we will get two new algebraic equations of the 4-th degree, which radicals are the golden proportion:

 (6) (7)

The analysis of the equation (7) results us to unexpected outcome, because this equation describes the power condition of the butadiene, the valuable chemical agent used for rubber production. The famous physicist Feynman has expressed his delight concerning the equation (7) in the following words: "What miracles exist in mathematics! According to my theory the golden proportion of the ancient Greeks gives the minimal power condition of the butadiene particle".

This fact at once increases our interest in the equations of the golden proportion of the higher degrees. These equations can be obtained, if we sequentially will consider the equations of the following type As an example it is possible to consider the following equations of the higher degrees:

The analysis of these equations demonstrates, that the numerical constants in the right-hand part of these equations are Fibonacci numbers, which would be considered by us we below. Generally the algebraic "golden" equations of the n-th degree is expressed as the following:

 (8)

where are Fibonacci numbers.

Let's note once again, that the main mathematical property of the equations (8) is that all of them have the common radical, the golden proportion.