arXiv subjects
Publications and source records attributed to Luis H. Gallardo.
The paper is about a polynomial variant of a classical arithmetic problem. More precisely, we obtain all non-splitting bi-unitary perfect polynomials over the prime field of two elements, which have only Mersenne polynomials as odd irreducible divisors.
We study an analogue of a classical arithmetic problem over the ring of polynomials. We prove that $m = 5$ is the minimal number such that the sums of any two distinct polynomials in a set of $m$ polynomials over $\F_2[x]$ cannot all be of the form $x^k(x+1)^{\ell}$.
We study a natural analogue of Collatz's Conjecture for polynomials over $\mathbb{F}_2$.
We provide new sufficient conditions under which Ryser's conjecture holds.
In this note, we fix a gap in a proof of the first author that 28 is the only even perfect number which is the sum of two perfect cubes. We also discuss the situation for higher powers.
We describe combinatorial properties of the defining row of a circulant Hadamard matrix by exploiting its orthogonality to subsequent rows, and show how to exclude several particular forms of these matrices.
Assume that $H$ is a circulant Hadamard matrix of order $n\geq 4$. We consider an appropriate stochastic matrix $S$ of order $n$ depending on $H$. This allows us to prove that $n = 4$. Thus, there are only $10$ circulant Hadamard matrices.
We give all splitting bi-unitary perfect polynomials over the field $\mathbb{F}_4$ and some splitting ones over $\mathbb{F}_{p^2}$, if $p$ is an odd prime.
We build a variant of Collatz Conjecture for polynomials over $\mathbb{F}_2$ and we prove that it is solved. By the way, we give several examples.
We give necessary conditions for perfection of some families of odd numbers with special multiplicative forms. Extending earlier work of Steuerwald, Kanold, McDaniel et al.
We work an analogue of a classical arithmetic problem over polynomials. More precisely, we study the fixed points $F$ of the sum of divisors function $σ: F_2[x] \mapsto F_2[x]$ (defined \emph{mutatis mutandi} like the usual sum of divisors over the integers) of the form $F := A^2 \cdot S$, $S$ square-free, with $ω(S) \leq 3$, coprime with $A$, for $A$ even, of whatever degree, under some conditions. This gives a characterization of $5$ of the $11$ known fixed points of $σ$ in $F_2[x]$
We adapt (over $\mathbb{F}_2$) the general notions of multiplicative function, Dirichlet convolution and Inverse. We get some interesting results, namely necessary conditions for an odd binary polynomial to be perfect. Note that we are inspired by the "analogous" works in \cite{Gall-Rahav-newcongr} and \cite{Touchard}, about odd perfect numbers.
The paper is about an arithmetic problem in $\F_2[x]$. We give \emph{admissible} (necessary) conditions satisfied by a set of odd prime divisors of perfect polynomials over $\F_2$. This allows us to prove a new characterization of \emph{all} known perfect polynomials, and to open a way of finding more of them (if they exist).
We address an arithmetic problem in the ring $\F_2[x]$ related to the fixed points of the sum of divisors function. We study some binary polynomials $A$ such that $σ(A)/A $ is still a binary polynomial. Technically, we prove that the only (unitary) perfect polynomials over $\F_2$ that are products of $x$, $x+1$ and of Mersenne primes are precisely the nine (resp. nine "classes") known ones. This follows from a new result about the factorization of $M^{2h+1} +1$, for a Mersenne prime $M$ and for a positive integer $h$.
We get some results about the factorization of $ϕ_p(M) \in {\mathbb{F}}_2[x]$, where $p$ is a prime number, $ϕ_p$ is the corresponding cyclotomic polynomial and $M$ is a Mersenne prime (polynomial). By the way, we better understand the factorization of the sum of the divisors of $M^{2h}$, for a positive integer $h$.
The only (unitary) perfect polynomials over $\mathbb{F}_2$ that are products of $x$, $x+1$ and Mersenne primes are precisely the nine (resp. nine "classes") known ones. This follows from a new result about the factorization of $M^{2h+1} +1$, for a Mersenne prime $M$ and for a positive integer $h$. Other consequences of such a factorization are new results about odd perfect polynomials.
We prove that there is no circulant Hadamard matrix $H$ with first row $[h_{1},\ldots,h_{n}]$ of order $n>4$, under a condition about a sum of scalar products of rows of two other circulant matrices of size $n/2$ associated to $H.$
For every positive integer $k$ such that $k>1,$ there are an infinity of odd integers $h$ with $ω(h) =k$ distinct prime divisors such that there do not exist a Circulant Hadamard matrix $H$ of order $n=4h^2.$ Moreover, our main result implies that for all of the odd numbers $h$, with $1< h < 10^{13}$ there is no Circulant Hadamard matrix of order $n=4h^2.$