Haissam Chehade

Let k be a positive integer. A polynomial A∈F_2[x] is called k-unitary perfect if the sum of the k-th powers of its distinct unitary divisors equals A^k. In this paper, we focus on the case k=2^n and prove that every 2^n-unitary perfect polynomial over F_2 is necessarily even. Moreover, we obtain a complete classification of all even 2^n-unitary perfect polynomials having at most three distinct irreducible factors. In particular, we characterize all such polynomials of the formA=x^a (x+1)^b P^h, where P is a Mersenne prime over F_2 and a, b, and h are positive integers. As a consequence, several explicit infinite families of k-unitary perfect polynomials over F_2 are obtained.

A Mersenne number is defined as a number of the form M_n=2^n-1, where n is a positive integer. The first five Mersenne numbers are 1, 3, 7, 15, and 31. A divisor d of a positive integer m=p^k, where p is a prime, is termed an exponential divisor if it satisfies d=p^t with t dividing k, and it is denoted as d|_e m. Two integers a and b share a common exponential divisor if they have the same prime factors. The greatest common exponential divisor (GCED) of two integers a and b is denoted by gced(a, b). A set S is called exponential factor-closed if the exponential divisor of every element of S also belongs to S. Similarly, S is GCED-closed if gced(a, b) belongs to S for every pair a,b in S. If S is an exponential factor-closed set of distinct positive integers arranged in increasing order, the GCED matrix associated with S is the matrix M, where each entry M_ij is given by gced(a_i,a_j). The Mersenne GCED matrix M associated with S is a square matrix where each entry M_ij is of the form gced(2^(a_i )-〖1,2〗^(a_j )-1). This paper introduces the concept of Mersenne GCED square matrices defined on a non-exponential factor-closed set. We establish a comprehensive characterization of their fundamental properties, including their structure, determinant, reciprocal, and inverse.

In this paper, we extend the concept of lcem matrices beyond the classical domain of natural integers into the domain of unique factorization domains. We investigate the structure of these matrix types when applied to both arbitrary sets and gced-closed sets. Furthermore, we find the determinant, the trace and the inverse of such matrices. To simplify these ideas, we employ domains such as the Gaussian integers domain and the domain of polynomials defined over finite fields.

In this paper, we give all non splitting bi-unitary superperfect polynomials divisible by one or two irreducible polynomials over the prime field of two elements. We prove the nonexistence of odd bi-unitary superperfect polynomials over F2.

In this article, a new deterministic primality test for Mersenne primes is presented. It also includes a comparative study between well-known primality tests in order to identify the best test. Moreover, new modifications are suggested in order to eliminate pseudoprimes. The study covers random primes such as Mersenne primes and Proth primes. Finally, these tests are arranged from the best to the worst according to strength, speed, and effectiveness based on the results obtained through programs prepared and operated by Mathematica, and the results are presented through tables and graphs.

An extension of the GCED matrices from the domain of natural integers to the unique factorization domain is given. The structure of these type of matrices defined on both arbitrary sets and GCED-closed sets are presented. Moreover, we present exact expressions for the determinant and the inverse of such matrices. The domains of Gaussian integers and polynomials over finite fields are used to illustrate the work.

In this paper, we use a generalized form for the Jordan totient function in order to extend the Reciprocal power GCDQ matrices and power LCMQ matrices from the standard domain of natural integers to Euclidean domains. Structural theorems and determinantal arguments defined on both arbitrary and factor-closed q-ordered sets are presented over such domains. We illustrate our work in the case of Gaussian integers.