An elementary heuristic for Hardy-Littlewood extended Goldbach's conjecture

TL;DR

This paper presents an elementary combinatorial heuristic, based on probability models and concentration bounds, to predict the extended Goldbach's conjecture, avoiding complex analysis and building on existing additive prime number theory heuristics.

Contribution

It introduces a new elementary heuristic model for the extended Goldbach's conjecture using hypergeometric distributions and concentration inequalities, differing from traditional complex-analytic approaches.

Findings

Heuristic predicts the validity of the extended Goldbach's conjecture.

Model aligns with known features of additive prime number heuristics.

Provides a probabilistic framework for understanding prime sums.

Abstract

The goal of this paper is to describe an elementary combinatorial heuristic that predicts Hardy and Littlewood's extended Goldbach's conjecture. We examine common features of other heuristics in additive prime number theory, such as Cram\'er's model and density-type arguments, both of which our heuristic draws from. Apart from the prime number theorem, our argument is entirely elementary, in the sense of not involving complex analysis. The idea is to model sums of two primes by a hypergeometric probability distribution, and then draw heuristic conclusions from its concentration behavior, which follows from Hoeffding-type bounds.