Solution of single parameter Bring quintic equation

TL;DR

This paper introduces a novel method to solve the Bring quintic equation by transforming it into an infinite series and then into a quartic, enabling solutions for cases where |a| > 1.

Contribution

The paper presents a new approach that converts the quintic into a convergent series and a quartic, introducing ultraradicals for solving the equation.

Findings

Method converges for |a| > 1

Transforms quintic into a quartic with ultraradicals

Provides explicit real solutions for the equation

Abstract

In this paper, we propose a new method to obtain a solution to a single-parameter Bring quintic equation of the form, $x^{5}+x=a$, where $a$ is real. The method transforms the given quintic equation to an infinite but convergent series expression in $(x/a)$, which is further transformed to a quartic equation in a novel fashion. The coefficients of the quartic equation so obtained are some kind of infinite series expressions in $a^{-4}$, which are termed as \textit{ultraradicals}. The quartic equation is then solved and its one real solution is picked; further using this, the real solution of quintic equation, $x^{5}+x=a,$ is extracted. The ultraradicals used in this method converge for $|a| > 1$; hence the method can be used when $|a| > 1$.