Zhi-Wei Sun's 1-3-5 Conjecture and Variations
Number Theory
2025-08-07 v3
Abstract
In this paper, using quaternion arithmetic in the ring of Lipschitz integers, we present a proof of Zh\`i-W\v{e}i S\={u}n's "1-3-5 conjecture" for integral solutions, and for all natural numbers greater than a specific constant. This, together with computations done by the authors and a colleague, which checked the validity of the conjecture up to that constant, completely proves the 1-3-5 conjecture. We also establish some variations of this conjecture.
Cite
@article{arxiv.2003.02592,
title = {Zhi-Wei Sun's 1-3-5 Conjecture and Variations},
author = {António Machiavelo and Nikolaos Tsopanidis},
journal= {arXiv preprint arXiv:2003.02592},
year = {2025}
}
Comments
This is a replacement that corrects, in an appendix at the end (authored by Jorge Costa, Ant\'onio Machiavelo and Rog\'erio Reis), an error on the original version, that has been published in the Journal of Number Theory, volume 222, 2021, pp. 1--20