6 papers
Metaconjugation and Quadratic Forms
Adriana Cardoso, António Leite, António Machiavelo +2
We present a quaternionic proof that the quadratic form represents all positive integers, and that the form represents all natural numbers…
The Dedekind-Hasse Criterion in Quaternion Algebras
Adriana Cardoso, António Machiavelo, António Machiavelo
We show that a criterion for an integral domain to be a principal ideal domain (PID), due to Dedekind and Hasse, can also be applied in quaternion orders, and that it can be used t…
On supratopologies, normalized families and Frankl conjecture
André Carvalho, António Machiavelo
We introduce some generalized topological concepts to deal with union-closed families, and show that one can reduce the proof of Frankl's conjecture to some families of so-called s…
Zhi-Wei Sun's 1-3-5 Conjecture and Variations
António Machiavelo, Nikolaos Tsopanidis
In this paper, using quaternion arithmetic in the ring of Lipschitz integers, we present a proof of Zhì-WÄi SÅ«n's "1-3-5 conjecture" for integral solutions, and for all natural…
Some Connections Between The Arithmetic and The Geometry of Lipschitz Integers
António Machiavelo, LuÃs Roçadas
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 Lipsch…
On the Cycle Structure of the Metacommutation Map
António Leite, António Machiavelo
Cohn and Kumar showed that the permutation on the set of the classes of left associated Hurwitz primes above an odd prime induced through metacommutation by a Hurwitz prime $ξ…