  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 all
integers, 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.
