Uwe Kraeft Libri

In the theory of commutative rings and their ideals, you can find mayor and not surpassed applications in number theory, especially in factorisation of numbers in quadratic and higher number fields, where roots are adjunctions to rational numbers and real approximations are notneeded. In this text, you can find in 10 chapters homomorphisms, rings, and ideals in elementary algebra, fundamental algebra and examples of rings, polynomial rings, residue classes, modules and ideals, primes, prime elements, and factorisation, residue classes and ideals, special rings, ideals in number theory, and a short history of ideals. After the text, a choke of literature, the, until now, collected corrections, and an index can be found.

In addition to the primary disciplines in number theory, numerous riddles, problems, and conjectures present both challenges and intrigue. These tasks engage not only professional mathematicians but also enthusiasts, leading to significant discoveries beyond mere entertainment. This text comprises ten chapters filled with examples of games, "magic" squares, riddles, secrets, and paradoxes. It explores the Ulam spiral, the Collatz conjecture, Catalan's conjecture/Mihăilescu's theorem, and Fermat's Last Theorem, alongside other renowned problems. The discussion also covers various conjectures related to primes, infinity, and geometry, including Zeno's Paradox and the challenge of squaring the circle, as well as the concept of vector spaces. Through these topics, the text illustrates the depth and complexity of number theory while inviting readers to engage with its many puzzles and enigmas.

This text is mainly composed of contributions in several former books of the author, which are discussed more detailed here. As the title points up, the aim is the search for structures of primes and their relations to Goldbach's Conjecture, which says that always (odd) primes PI and P2 exist which are solutions to the Diophantine Equation 2n = PI +P2 with natural numbers n>2. In this text, you can find in 8 chapters, after an introduction, special cases of Goldbach's Conjecture, basic formulae of primes, factorial numbers, structures of composite numbers and Goldbach's Conjecture, characteristics of equal sums of primes or composite numbers, the sequence of primes I or II, and additional strategies of proofs.

The text fills a last greater gap in the sequence of the author's basic studies in number theory. Integers with or without special characteristics can be represented by integer sum functions of one or more natural numbers, which may be primes or composite numbers. In only few cases, there can be direct methods for finding solutions to equations or tests for the possibility of solutions. Within the latter, congruences of numbers or functions can be regarded. In seven chapters, binary quadratic forms of two variables, general quadratic forms and algorithms, remarks about general forms, theoretical and practical representations of natural numbers, some useful congruences, and Goldbach's Conjecture are discussed. After the text, a choice of literature, collected corrections to former books of the author, and a complete index are given.

In this book, which shows besides definitions a choice of basic theorems, corollaries, conjectures, and formulae which are important in number theory, you will find no proofs. For better understanding, in some cases, also special theorems are given in addition to the most general ones. In reading a chapter, you should be able to get in a few hours first information, find a base for further developments, or repeat known facts. The choice of theorems and formulae is arbitrary and strictly focused on number theory. The main source is the author's sequence of studies in number theory. You will find more information in the selected literature at the end of the book.