Abstract

The special case of the (p+1)th degree algebraic equations of the kind x^p+1 = x^p + 1 (p = 1,2,3,...) is researched in the present article. For the case p = 1, the given equation is reduced to the well-known Golden Proportion equation x² = x + 1. These equations are called the golden algebraic equations because the golden p-proportions τ_p, special irrational numbers that follow from Pascal's triangle, are their roots. A research on the general properties of the roots of the golden algebraic equations is carried out in this article. In particular, formulas are derived for the golden algebraic equations that have degree greater than p + 1. There is reason to suppose that algebraic equations derived by the authors in the present article will interest theoretical physicists. For example, these algebraic equations could be found in the research of the energy relationships within the structures of many compounds and physical particles. For the case of butadiene (C₄H₆), this fact is proved by the famous physicist Richard Feynman.

1. Introduction

The solutions of algebraic equations have long attracted the special attention of mathematicians. As a result, interest in these important mathematical problems promoted the development of algebra. Ancient mathematicians discovered the formulas for the roots of the first and second degree algebraic equations (polynomials). However, mathematicians did not develop formulas for the roots of the third and fourth degree algebraic equations until the 16th and 17th centuries. The general formulas for the roots of polynomials of fifth degree and higher does not exist. Solutions of the algebraic equations involving radicals led Galois, the best-known French mathematician of the 19th century, to the general theory of algebraic equations named Galois Theory.

Among the infinite set of all algebraic equations, the following second degree algebraic equation is of special interest:

x2=x+1.

(1)

The famous Golden Proportion found often in nature, is the root of this equation. Many objects alive in the

natural world that possess pentagonal symmetry, such as marine stars, inflorescences of many flowers, and phyllotaxis objects have a numerical description given by the Fibonacci numbers which are themselves based on the Golden Proportion. In the last few years, the Golden Proportion has played an increasing role in modern physical research [1-14].

Let us consider the following algebraic equation with a degree (p + 1) introduced in [15]:

xp+1 = xp + 1,

(2)

where p takes its values from the set {0,1,2,3,...}.

As shown in [15-20], the positive roots of Eq. (2) generate an important class of the irrational numbers τ_p, where p = {0,1,2,3,...}. The numbers, τ_p, express some deep mathematical properties of the Pascal's triangle.

Note that the golden proportion equation (1) is a special case (p = 1) of the more general equation (2). This fact enables us to name the roots τ_p of the general equation (2) as the generalized golden proportions or the golden p-propor-tions [15]. Further, we designate Eq. (2) to be the golden basic algebraic equation with degree (p + 1).

The golden basic algebraic equation (1) is the simplest one. Note that Eq. (1) is so simple that a student can solve it, i.e., find its roots. However, even this elementary algebraic equation generates the following non-trivial question: Are there algebraic equations of higher degree with roots that are the golden proportion? It is possible to address this question with Eq. (2) for a given p О {1,2,3,...}. That is, we may use Eq. (2) in order to discover whether there exist golden algebraic equations of degree greater than p + 1 which have the golden p-proportion τ_p as their root. The search for answers to these and other questions concerning the golden algebraic equation given by (2) is the purpose of this article.


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