Almost all primes are not needed in Ternary Goldbach

TL;DR

This paper constructs a special set of primes with restrictive properties, showing that the ternary Goldbach conjecture remains true even when only using these primes, highlighting the conjecture's robustness.

Contribution

It introduces a novel set of primes based on admissible congruences, demonstrating the conjecture's validity with a significantly restricted prime subset.

Findings

Almost all primes are excluded from the set while maintaining the conjecture's truth.

The ternary Goldbach conjecture holds with primes restricted to the constructed set.

The set of primes is defined via an expanding system of admissible congruences.

Abstract

The ternary Goldbach conjecture states that every odd number $m \geqslant 7$ can be written as the sum of three primes. We construct a set of primes $\mathbb{P}$ defined by an expanding system of admissible congruences such that almost all primes are not in $\mathbb{P}$ and still, the ternary Goldbach conjecture holds true with primes restricted to $\mathbb{P}$.