All bi-unitary perfect polynomials over $\F_2$ with at most four irreducible factors
We give, in this paper, all bi-unitary perfect polynomials over the prime field $\F_2$, with at most four irreducible factors.
math.NT↗
arXiv subjects
Olivier Rahavandrainy
Publications and source records attributed to Olivier Rahavandrainy.
All bi-unitary perfect polynomials over $\F_2$ with only Mersenne primes as odd divisors
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.
math.NT↗
A variant of Collatz's Conjecture over Binary Polynomials
We study a natural analogue of Collatz's Conjecture for polynomials over $\mathbb{F}_2$.
math.NT↗
Non-splitting bi-unitary perfect polynomials over $\mathbb{F}_4$ with less than five prime factors
We identify all non-splitting bi-unitary perfect polynomials over the field $\mathbb{F}_4$, which admit at most four irreducible divisors. There is an infinite number of such divisors.
math.NT↗
Combinatorial properties of Circulant Hadamard matrices
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.
math.CO↗
On splitting bi-unitary perfect polynomials over $\mathbb{F}_{p^2}$
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.
math.NT↗
On Collatz Conjecture for binary polynomials
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.
math.NT↗
Some multiplicative functions over $\mathbb{F}_2$
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.
math.NT↗
On bi-unitary perfect polynomials over $F_2$
We give all bi-unitary non splitting even perfect polynomials over the prime field of two elements, which are divisible by Mersenne irreducible polynomials raised to special exponents. We also identify all bi-unitary perfect polynomials over the same field, with at most four irreducible factors. We then complete, in this manner, a list given by J.T. B. Beard Jr.
math.NT↗
Admissible family for binary perfect polynomials
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).
math.NT↗
All even (unitary) perfect polynomials over $\F_2$ with only Mersenne primes as odd divisors
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 $h$.
math.NT↗
Factorization of cyclotomic polynomial values at Mersenne primes
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 $h$.
math.NT↗
On (unitary) perfect polynomials over $\mathbb{F}_2$ with only Mersenne primes as odd divisors
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.
math.NT↗
All unitary perfect polynomials over $F_2$ with less than five distinct prime factors
We find all unitary perfect polynomials over the prime field $\F_2$ with less than five distinct prime factors.
math.NT↗
On splitting perfect polynomials over $\mathbb{F}_{p^2}$
We study some properties of the exponents of the terms appearing in the splitting perfect polynomials over $\mathbb{F}_{p^2}$, where $p$ is a prime number. This generalizes the work of Beard et al. over $\mathbb{F}_p$. Corrected paper. Older Lemmas 2.17 to 2.20 in published version required a fixing done herein.
math.NT↗
Even perfect polynomials over $F_2$ with four prime factors
A perfect polynomial over the binary field $\F_2$ is a polynomial $A \in \F_2[x]$ that equals the sum of all its divisors. If $\gcd(A,x^2-x) \neq 1$ then we call $A$ even. The list of all even perfect polynomials over $\F_2$ with at most 3 prime factors in known. The object of this paper is to give the list of all even perfect polynomials over $\F_2$ with four prime factors. These are all the known perfect polynomials with four prime factors over $\F_2$.
math.NT↗