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by Lukasz Matysiak (Author)
In this publication, which is the result of many years of research, the author presents an innovative approach to square-free and radical factorizations, which is a generalization of classical factorizations. The problem of factoring numbers into prime factors is familiar to many, but often this problem is complex and depends on the number structure or other more advanced algebraic structures. The results in this publication will usually refer to the structure of monoids, and therefore in particular to classical integers. The author introduces certain types of factorization and his own definition of a square-free ideal, which is an analogy to ideals in a classic abstract algebra course. At the end of the topic of factorization, the author adds some results regarding the square-free numbers themselves, which have something to do with the famous Riemann hypothesis, which has not yet been solved. The author, moreover, presents results regarding the over a hundred-year-old hypotheses of Wieferich, Artin and Dirichlet regarding quadratic congruences in the world of number theory. This book also includes chapters on cryptology, including generalizations of the RSA cipher and the Diffie-Hellman protocol, which can have a significant impact on data security in the digital age. The author also proposes new algebraic cryptographic systems, which, in his opinion, have the potential to become an important tool in information protection. Further in the publication, the reader will find proposals for securing the cost and production functions, which may be of significant importance for industry and economics. Finally, the author presents a mathematical model that improves the preference theory by analyzing the differences between individual and social preferences, which may contribute to a better understanding of consumer behavior. This publication is intended for mathematicians, cryptologists, economists, as well as students and scientists interested in the latest achievements in the field of applied mathematics. This book is not only a collection of scientific results, but also a source of inspiration for future research in the field of mathematics and its applications.
