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 Zhi-Wei Sun's “1-3-5 conjecture” for all integers, and reduce the general case to its verification up to 1.052 × 10 11 . The computational verification was performed by the authors and a colleague, concluding the proof of Sun's 1-3-5 conjecture. We also establish some variations of this conjecture.