The “golden” algebraic equations

Abstract

Abstract The special case of the (p + 1)th degree algebraic equations of the kind xp+1 = xp + 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 τp, 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.