Related papers: Some multiplicative functions over $\mathbb{F}_2$
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 give necessary conditions satisfied by the set of odd prime divisors of binary perfect polynomials. This allows us to get a new characterization of all the known perfect binary 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…
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}…
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…
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 study exponential sums whose coefficients are completely multiplicative and belong to the complex unit disc. Our main result shows that such a sum has substantial cancellation unless the coefficient function is essentially a Dirichlet…
In this paper we present a new class of perfect nonlinear %Dembowski-Ostrom polynomials over $\mathbb{F}_{p^{2k}}$ for any odd prime $p$. In addition, we show that the new perfect nonlinear functions are CCZ-inequivalent to all the…
Numerous results on self-reciprocal polynomials over finite fields have been studied. In this paper we generalize some of these to a-self reciprocal polynomials defined in [4]. We consider some properties of the divisibility of a-reciprocal…
We provide some conditions for the graph of a Hoelder-continuous function on \bar{D}, where \bar{D} is a closed disc in the complex plane, to be polynomially convex. Almost all sufficient conditions known to date --- provided the function…
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.…
In this paper we consider in detail the composition of an irreducible polynomial with X^2 and suggest a recurrent construction of irreducible polynomials of fixed degree over finite fields of odd characteristics. More precisely, given an…
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 $\sigma(A)/A $ is still a binary polynomial. Technically, we prove that the…
Let $p>3$ be a prime. We show that, for each integer $d$ with $p \leq d \leq 2(p-1)$, there exists a generalized almost perfect nonlinear (GAPN) binomial or trinomial over $\mathbb{F}_{p^2}$ of algebraic degree $d$. We start by deriving…
Much recent progress has been made concerning the probable existence of Odd Perfect Numbers, forming part of what has come to be known as Sylvester's Web Of Conditions. This paper proves some results concerning certain properties of the…
We present a sufficient condition for irreducibility of forcing algebras and study the (non)-reducedness phenomenon. Furthermore, we prove a criterion for normality for forcing algebras over a polynomial base ring with coefficients in a…
A class of bilinear permutation polynomials over a finite field of characteristic 2 was constructed in a recursive manner recently which involved some other constructions as special cases. We determine the compositional inverses of them…
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…
We shall given a new effectively computable upper bound of odd perfect numbers whose Euler factors are powers of fixed exponent, improving our old result in T. Yamada, Colloq. Math. 103 (2005), 303--307.
We give some classes of power maps with low $c$-differential uniformity over finite fields of odd characteristic, {for $c=-1$}. Moreover, we give a necessary and sufficient condition for a linearized polynomial to be a perfect $c$-nonlinear…