Understanding Anatomy of Exponential Diophantine 3-term Equations

This book develops a classification structure to classify solutions to exponential diophantine 3-term equations. This classification is utilized in understanding of various equations starting from A + B = C to the Fermat's equation and the Beal equation. It is shown that for Fermat's equation, non-primitive solutions cannot exist unless primitive solutions exist. For Beal's equation, non-primitive solutions can exist even if primitive solutions may not exist.

This monograph is specially written for students and researchers in mathematics, particularly number theory. Professors and instructors teaching this subject will find it very useful. This monograph develops a clear understanding of the structure of the exponential diophantine 3-term equations and their solutions. Various components of an exponential diophantine equation and their roles are described in detail. A combinatorial approach is used to classify and analyze solutions to exponential diophantine 3-term equations. To classify these solutions, all possible ways these solutions can occur are examined. The analysis presented reveals that these solutions can be classified into three classes and six sub-classes. Properties of each class of solutions are described in detail. We observe that each exponential diophantine 3-term equation imposes its own requirements on its solutions. We discuss how these requirements of an exponential diophantine equation can affect solutions in each class. We apply the classification developed here to relate to the results for the Pythagorean equation, the Catalan equation, and the Ramanujan-Nagell equation. Finally, we employ the concepts developed in this monograph and present twelve proofs and four conjectures that help us analyze and understand the Fermat's Last Theorem and the Beal conjecture. These four conjectures are much smaller in scope.

Rajen Merchant studied at Bombay University for one year and then graduated as Bachelor of Technology (a five year course) at the Indian Institute of Technology, Bombay, India. After completing his Master of Science at Carnegie-Mellon University, Pittsburgh, USA, he migrated to Canada. He obtained his Master of Business Administration degree from Queen's University, Kingston, Canada. He has a keen interest in mathematics and its applications. His pursuit for mathematical accuracy has helped him better understand the intricacies of the exponential Diophantine equations as well as the elliptical co-ordinate system. This is his third monograph. The first monograph, titled "Understanding the Elastic Stress Field Around an Elliptical Hole in a Thin Plate (in the deformed configuration)" – ISBN 978-1-48359-262-6, was published in January 2017. The second monograph, titled "Understanding Non-Linear Strain-Displacement Relations in the Elliptical Co-ordinate System (with applications to holes in a thin plate)" - ISBN 978-1-54398-916-8, was published in December 2019. He intends to publish the fourth book soon. This fourth book will examine applications of the concepts presented in the first two monographs.