Journal of Advances in Mathematics and Computer Science <p style="text-align: justify;"><strong>Journal of Advances in Mathematics and Computer Science (ISSN:&nbsp;2456-9968) </strong>aims to publish original research articles, review articles and short communications, in all areas of mathematics and computer science. Subject matters cover pure and applied mathematics, mathematical foundations, statistics and game theory, use of mathematics in natural science, engineering, medicine, and the social sciences, theoretical computer science, algorithms and data structures, computer elements and system architecture, programming languages and compilers, concurrent, parallel and distributed systems, telecommunication and networking, software engineering, computer graphics, scientific computing, database management, computational science, artificial Intelligence, human-computer interactions, etc. This is a quality controlled, OPEN peer reviewed, open access INTERNATIONAL journal.</p> SCIENCEDOMAIN international en-US Journal of Advances in Mathematics and Computer Science 2456-9968 Fermat’s Last Theorem and Related Problems <p>Empirical evidence in support of generalizations of Fermat’s equation is presented. The empirical evidence consists mainly of results for the p = 3 case where Fermat’s Last Theorem is almost false. The empirical evidence also consists of results for general p values. The \pth power with respect to" concept (involving congruences) is introduced and used to derive these generalizations. The classical Furtw¨angler theorems are reformulated. Hasse used one of his reciprocity laws to give a more systematic proof of Furtw¨angler’s theorems. Hasse’s reciprocity law is modified to deal with a certain condition. Vandiver’s theorem is reformulated and generalized. The eigenvalues of 2p x 2p matrices for the p = 3 case are investigated. (There is a relationship between the modularity theorem and a re-interpretation of the quadratic reciprocity theorem as a system of eigenvalues on a finite-dimensional complex vector space.) A generalization involving generators and \reciprocity" has solutions for every p value.</p> Darell Cox Sourangshu Ghosh Eldar Sultanow ##submission.copyrightStatement## 2021-06-12 2021-06-12 6 34 10.9734/jamcs/2021/v36i530361 On the Ideal Based Zero Divisor Graphs of Unital Commutative Rings and Galois Ring Module Idealizations <p>Let R be a commutative ring with identity 1 and I is an ideal of R. The zero divisor graph of the ring with respect to ideal has vertices defined as follows: {u ∈ Ic | uv ∈ I for some v ∈ Ic}, where Ic is the complement of I and two distinct vertices are adjacent if and only if their product lies in the ideal. In this note, we investigate the conditions under which the zero divisor graph of the ring with respect to the ideal coincides with the zero divisor graph of the ring modulo the ideal. We also consider a case of Galois ring module idealization and investigate its ideal based zero divisor graph.</p> Owino Maurice Oduor ##submission.copyrightStatement## 2021-06-12 2021-06-12 1 5 10.9734/jamcs/2021/v36i530360