Fermat’s Last Theorem and Related Problems

2021-06-12T04:17:58+00:00

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.

On the Ideal Based Zero Divisor Graphs of Unital Commutative Rings and Galois Ring Module Idealizations

2021-06-12T04:04:04+00:00

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.

Journal of Advances in Mathematics and Computer Science

Fermat’s Last Theorem and Related Problems

On the Ideal Based Zero Divisor Graphs of Unital Commutative Rings and Galois Ring Module Idealizations