Report on Zhi-Wei Sun's "1-3-5 conjecture" and some of its refinements

TL;DR

This paper verifies a refined version of Zhi-Wei Sun's 1-3-5 conjecture for all natural numbers up to a large bound, completing its proof through computational methods and refinements.

Contribution

It provides the computational verification of a refined conjecture and completes the proof of Sun's 1-3-5 conjecture.

Findings

Verification up to 105,103,560,126 natural numbers

Completion of the proof of Sun's 1-3-5 conjecture

Refinement of the original conjecture

Abstract

We report here on the computational verification of a refinement of Zhi-Wei Sun's "1-3-5 conjecture" for all natural numbers up to 105 103 560 126. This, together with a result of two of the authors, completes the proof of that conjecture.