Zhi-Wei Sun's 1-3-5 Conjecture and Variations

Abstract

In this paper, using quaternion arithmetic in the ring of Lipschitz integers, we present a proof of Zhì-Wěi Sūn'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.

This is a replacement that corrects, in an appendix at the end (authored by Jorge Costa, António Machiavelo and Rogério Reis), an error on the original version, that has been published in the Journal of Number Theory, volume 222, 2021, pp. 1--20