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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 allintegers, and for all natural numbers greater than a specific constant. This,together with computations, done by the authors and a colleague, which checkedthe validity of the conjecture up to that constant, completely proves the 1-3-5conjecture. We also establish some variations of this conjecture.

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