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

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.