Ramanujan, the taxicab problem for polynomials, and the abc-conjecture

TL;DR

This paper explores polynomial equations related to Ramanujan's taxicab problem and their connection to the abc-conjecture, focusing on solvability conditions.

Contribution

It extends the classical taxicab problem to polynomial equations and investigates their solvability in the context of the abc-conjecture.

Findings

Analyzes solvability of polynomial equations similar to Ramanujan's taxicab problem.

Connects polynomial equations to the abc-conjecture.

Provides conditions under which these polynomial equations are solvable.

Abstract

Starting with Ramanujan's famous taxicab problem, we can study the solvability of the equations $p^n+q^n=r^n+s^n$ and, more generally, $p_1^{k_1}+\dots+p_m^{k_m}=0$ among polynomials.