Polynomials whose reducibility is related to the Goldbach conjecture

TL;DR

This paper introduces polynomials linked to the Goldbach conjecture, exploring their properties and proposing generalizations, with numerical evidence supporting their potential relevance to the conjecture.

Contribution

It presents a novel polynomial-based reformulation of the Goldbach conjecture and investigates their properties and possible extensions.

Findings

Polynomials $F_N$ relate to Goldbach's conjecture through divisibility properties.

Numerical evidence supports natural generalizations of these polynomials.

Asymptotic estimates on polynomial coefficients are provided.

Abstract

We introduce a collection of polynomials $F_N$, associated to each positive integer $N$, whose divisibility properties yield a reformulation of the Goldbach conjecture. While this reformulation certainly does not lead to a resolution of the conjecture, it does suggest two natural generalizations for which we provide some numerical evidence. As these polynomials $F_N$ are independently interesting, we further explore their basic properties, giving, among other things, asymptotic estimates on the growth of their coefficients.