arXiv subjects
Publications and source records attributed to António Leite.
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".
Cohn and Kumar showed that the permutation on the set of the classes of left associated Hurwitz primes above an odd prime $p$ induced through metacommutation by a Hurwitz prime $ξ$ of norm $q$ has either $0$, $1$ or $2$ fixed points, and that the permutation $τ_{ξ,p}$ induced on the non-fixed points splits into cycles of the same length. Here we show how to find the length of those cycles, in terms of $p$ and $ξ$, using cyclotomic polynomials over $\mathbb{F}_p$. We then show that, given an odd prime $p$, there is always a prime quaternion $ξ$ such that the permutation $τ_{ξ,p}$ has only one non-trivial cycle of length $p$. Finally, we give conditions for a prime $π$ of norm $p$ to be a fixed point of the aforementioned metacommutation map.