Important results surrounding the proof of Goldbach's ternary conjecture are presented in this book. Beginning with an historical perspective along with an overview of essential lemmas and theorems, this monograph moves on to a detailed proof of Vinogradov's theorem. The principles of the Hardy-Littlewood circle method are outlined and applied to Goldbach's ternary conjecture. New results due to H. Maier and the author on Vinogradov's theorem are proved under the assumption of the Riemann hypothesis. The final chapter discusses an approach to Goldbach's conjecture through theorems by L. G. Schnirelmann. This book concludes with an Appendix featuring a sketch of H. Helfgott's proof of Goldbach's ternary conjecture. The Appendix also presents some biographical remarks of mathematicians whose research has played a seminal role on the Goldbach ternary problem. The author's step-by-step approach makes this book accessible to those that have mastered classical number theory and fundamental notions of mathematical analysis. This book will be particularly useful to graduate students and mathematicians in analytic number theory, approximation theory as well as to researchers working on Goldbach's problem.

Important results surrounding the proof of Goldbach's ternary conjecture are presented in this book. Beginning with an historical perspective along with an overview of essential lemmas and theorems, this monograph moves on to a detailed proof of Vinogradov's theorem. The principles of the Hardy-Littlewood circle method are outlined and applied to Goldbach's ternary conjecture. New results due to H. Maier and the author on Vinogradov's theorem are proved under the assumption of the Riemann hypothesis. The final chapter discusses an approach to Goldbach's conjecture through theorems by L. G. Schnirelmann. This book concludes with an Appendix featuring a sketch of H. Helfgott's proof of Goldbach's ternary conjecture. The Appendix also presents some biographical remarks of mathematicians whose research has played a seminal role on the Goldbach ternary problem. The author's step-by-step approach makes this book accessible to those that have mastered classical number theory and fundamental notions of mathematical analysis. This book will be particularly useful to graduate students and mathematicians in analytic number theory, approximation theory as well as to researchers working on Goldbach's problem.

Uncle Petros is a family joke. An ageing recluse, he lives alone in a suburb of Athens, playing chess and tending to his garden. If you didn't know better, you'd surely think he was one of life's failures. But his young nephew suspects otherwise. For Uncle Petros, he discovers, was once a celebrated mathematician, brilliant and foolhardy enough to stake everything on solving a problem that had defied all attempts at proof for nearly three centuries - Goldbach's Conjecture. His quest brings him into contact with some of the century's greatest mathematicians, including the Indian prodigy Ramanujan and the young Alan Turing. But his struggle is lonely and single-minded, and by the end it has apparently destroyed his life. Until that is a final encounter with his nephew opens up to Petros, once more, the deep mysterious beauty of mathematics. Uncle Petros and Goldbach's Conjecture is an inspiring novel of intellectual adventure, proud genius, the exhilaration of pure mathematics - and the rivalry and antagonism which torment those who pursue impossible goals.

Second edition sold 2241 copies in N.A. and 1600 ROW. New edition contains 50 percent new material.

This book provides a detailed description of a most important unsolved mathematical problem OCo the Goldbach conjecture. Raised in 1742 in a letter from Goldbach to Euler, this conjecture attracted the attention of many mathematical geniuses. Several great achievements were made, but only until the 1920''s. The book gives an exposition of these results and their impact on mathematics, particularly, number theory. It also presents (partly or wholly) selections from important literature, so that readers can get a full picture of the conjecture."

This text investigates Waring's problem, approximation by fractional parts of the values of a polynomial, estimates for Weyl sums, distribution of fractional parts of polynomial values, Goldbach's problem, more. 1954 edition.

This book provides a detailed description of a most important unsolved mathematical problem — the Goldbach conjecture. Raised in 1742 in a letter from Goldbach to Euler, this conjecture attracted the attention of many mathematical geniuses. Several great achievements were made, but only until the 1920's. The book gives an exposition of these results and their impact on mathematics, particularly, number theory. It also presents (partly or wholly) selections from important literature, so that readers can get a full picture of the conjecture.

Upon publication, the first edition of the CRC Concise Encyclopedia of Mathematics received overwhelming accolades for its unparalleled scope, readability, and utility. It soon took its place among the top selling books in the history of Chapman & Hall/CRC, and its popularity continues unabated. Yet also unabated has been the d

In a manner accessible to beginning undergraduates, An Invitation to Modern Number Theory introduces many of the central problems, conjectures, results, and techniques of the field, such as the Riemann Hypothesis, Roth's Theorem, the Circle Method, and Random Matrix Theory. Showing how experiments are used to test conjectures and prove theorems, the book allows students to do original work on such problems, often using little more than calculus (though there are numerous remarks for those with deeper backgrounds). It shows students what number theory theorems are used for and what led to them and suggests problems for further research. Steven Miller and Ramin Takloo-Bighash introduce the problems and the computational skills required to numerically investigate them, providing background material (from probability to statistics to Fourier analysis) whenever necessary. They guide students through a variety of problems, ranging from basic number theory, cryptography, and Goldbach's Problem, to the algebraic structures of numbers and continued fractions, showing connections between these subjects and encouraging students to study them further. In addition, this is the first undergraduate book to explore Random Matrix Theory, which has recently become a powerful tool for predicting answers in number theory. Providing exercises, references to the background literature, and Web links to previous student research projects, An Invitation to Modern Number Theory can be used to teach a research seminar or a lecture class.