The ternary Goldbach conjecture is true

TL;DR

This paper proves the long-standing ternary Goldbach conjecture, demonstrating that every odd integer greater than 5 can be expressed as the sum of three primes, using advanced analytic number theory techniques.

Contribution

It introduces an optimized large sieve for primes and refines estimates on exponential sums, advancing the circle method approach for Goldbach's problem.

Findings

Proof of the ternary Goldbach conjecture

Improved estimates on exponential sums

Enhanced large sieve for primes

Abstract

The ternary Goldbach conjecture, or three-primes problem, asserts that every odd integer $n$ greater than $5$ is the sum of three primes. The present paper proves this conjecture. Both the ternary Goldbach conjecture and the binary, or strong, Goldbach conjecture had their origin in an exchange of letters between Euler and Goldbach in 1742. We will follow an approach based on the circle method, the large sieve and exponential sums. Some ideas coming from Hardy, Littlewood and Vinogradov are reinterpreted from a modern perspective. While all work here has to be explicit, the focus is on qualitative gains. The improved estimates on exponential sums are proven in the author's papers on major and minor arcs for Goldbach's problem. One of the highlights of the present paper is an optimized large sieve for primes. Its ideas get reapplied to the circle method to give an improved estimate for the minor-arc integral.