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.
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 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.
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