arXiv subjects
Publications and source records attributed to Luís Roçadas.
We present a quaternionic proof that the quadratic form $t^2+2x^2+5y^2+10z^2$ represents all positive integers, and that the form $t^2+x^2+7y^2+7z^2$ represents all natural numbers which are neither equal to $3 \cdot 7^\ell$ nor equal to $6 \cdot 7^\ell$, for any $\ell \in \mathbb{N}_0$. To do this we introduce a technique that may be useful for other purposes, and that we call "metaconjugation".
Some relationships between the arithmetic and the geometry of Lipschitz and Hurwitz integers are presented. In particular, it is shown that the (ternary) vector product of a Lipschitz integer $α$ with two other Lipschitz integers, both orthogonal to $α$, is a left and also a right multiple of $α$, and that the vector product of two left multiples of $α$ with any other Lipschitz integer is still a left multiple of $α$. We also provide new arithmetical proofs for some old results of Gordon Pall, and raise a geometric problem on the location of some integral quaternions that is related to the factorization of some integers.