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.

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.

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.

A divisor B of a nonzero polynomial A, defined over the prime field of two elements, is unitary (resp. bi-unitary) if $gcd(B,A/B)=1$ (resp. $gc{d}_u(B,A/B)=1)$, where $gc{d}_u(B,A/B)$ denotes the greatest common unitary divisor of B and $A/B$. We denote by ${\sigma }^{**}\left(A\right)$ the sum of all bi-unitary monic divisors of $A.$ A polynomial A is called a bi-unitary superperfect polynomial over ${\mathbb{F}}_2$ if the sum of all bi-unitary monic divisors of ${\sigma }^{**}\left(A\right)$ equals $A.$ In this paper, we give all bi-unitary superperfect polynomials divisible by one or two irreducible polynomials over ${\mathbb{F}}_2$. We prove the nonexistence of odd bi-unitary superperfect polynomials over ${\mathbb{F}}_2$.

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.

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.