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António Machiavelo

Publications and source records attributed to António Machiavelo.

12 recordsLinked to original sources
arXiv2026

The Dedekind-Hasse Criterion in Quaternion Algebras

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 to build a finite algorithm to determine if a given order is a principal left (or right) ideal domain. Using this algorithm, we give an alternative proof that the maximal orders of discriminant 7 and 13, which are non-Euclidean, are PIDs. We also provide a completely arithmetic proof of a result of Gordon Pall that shows that, in an order that is a PID, an element of whose norm is divisible by an integer $m$ always has a left and a right divisor with norm $m$. This easily yields the existence and uniqueness (up to associates) of factorizations of a quaternion modeled on a factorization of its norm.

math.NT↗
arXiv2026

Metaconjugation and Quadratic Forms

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".

math.NT↗
arXiv2025

On supratopologies, normalized families and Frankl conjecture

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 supratopological spaces. We prove some results on the structure of normalized families, presenting a new way of reducing such a family to a smaller one using dual families. Applying our reduction method, we prove a refinement of a conjecture originally proposed by Poonen. Finally, we show that Frankl's Conjecture holds for the class of families obtained from successively applying the reduction process to a power set.

math.CO↗
arXiv2025

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

In this paper, using quaternion arithmetic in the ring of Lipschitz integers, we present a proof of Zhì-Wěi Sūn's "1-3-5 conjecture" for integral solutions, 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.

math.NT↗
arXiv2025

Some Connections Between The Arithmetic and The Geometry of Lipschitz Integers

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.

math.NT↗
arXiv2025

On the Cycle Structure of the Metacommutation Map

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.

math.NT↗
arXiv2024

On Sum Graphs over Some Magmas

We consider the notions of sum graph and of relaxed sum graph over a magma, give several examples and results of these families of graphs over some natural magmas. We classify the cycles that are sum graphs for the magma of the subsets of a set with the operation of union, determine the abelian groups that provide a sum labelling of $C_4$, and show that $C_{4\ell}$ is a sum graph over the abelian group $\mathbb{Z}_f\times\mathbb{Z}_f$, where $f=f_{2\ell}$ is the corresponding Fibonacci number. For integral sum graphs, we give a linear upper bound for the radius of matchings, improving Harary's labelling for this family of graphs, and give the exact radius for the family of totally disconnected graphs. We found integer labellings for the 4D-cube, giving a negative answer to a question of Melnikov and Pyatikin, actually showing that the 4D-cube has infinitely many primitive labellings. We have also obtained some new results on mod sum graphs and relaxed sum graphs. Finally, we show that the direct product operation is closed for strong integral sum graphs.

math.CO↗
arXiv2021

On the Uniform Distribution of Regular Expressions

Although regular expressions do not correspond univocally to regular languages, it is still worthwhile to study their properties and algorithms. For the average case analysis one often relies on the uniform random generation using a specific grammar for regular expressions, that can represent regular languages with more or less redundancy. Generators that are uniform on the set of expressions are not necessarily uniform on the set of regular languages. Nevertheless, it is not straightforward that asymptotic estimates obtained by considering the whole set of regular expressions are different from those obtained using a more refined set that avoids some large class of equivalent expressions. In this paper we study a set of expressions that avoid a given absorbing pattern. It is shown that, although this set is significantly smaller than the standard one, the asymptotic average estimates for the size of the Glushkov automaton for these expressions does not differ from the standard case.

cs.FL↗
arXiv2015

Partial Derivative Automaton for Regular Expressions with Shuffle

We generalize the partial derivative automaton to regular expressions with shuffle and study its size in the worst and in the average case. The number of states of the partial derivative automata is in the worst case at most 2^m, where m is the number of letters in the expression, while asymptotically and on average it is no more than (4/3)^m.

cs.FL↗
arXiv2014

Statistical Study On The Number Of Injective Linear Finite Transducers

The notion of linear finite transducer (LFT) plays a crucial role in some cryptographic systems. In this paper we present a way to get an approximate value, by random sampling, for the number of non-equivalent injective LFTs. By introducing a recurrence relation to count canonical LFTs, we show how to estimate the percentage of $τ$-injective LFTs. Several experimental results are presented, which by themselves constitute an important step towards the evaluation of the key space of those systems.

cs.FL↗
arXiv2012

Dynamics of a quasi-quadratic map

We consider the map X defined on the rational numbers given by x --> x * ceil(x), where ceil(x) denotes the smallest integer greater than or equal to x, and study the problem of finding, for each rational, the smallest number of iterations of X that eventually sends it into an integer. Given two natural numbers M and n, we prove that the set of irreducible fractions with denominator M whose orbits by X reach an integer in exactly n iterations is a disjoint union of congruence classes modulo M^n, establishing along the way a finite procedure to ascertain them. We also describe an efficient algorithm to decide if an orbit fails to hit an integer until a prescribed number of iterations, and deduce that the probability that an orbit enters the set of integers is equal to one.

math.NT↗