In this book the authors have collected and systematically present solutions to problems of the analytic theory of numbers, introducing simplifications where possible.

Elementary methods in analytic number theory, (Rand McNally mathematics series)

Science & Math

Gelfond, A. O.

Linnik, Yu. V.

Mathematics

Rand-McNally

Primarily an advanced study of the modern theory of transcendental and algebraic numbers, this treatment by a distinguished Soviet mathematician focuses on the theory's fundamental methods. The text also chronicles the historical development of the theory's methods and explores the connections with other problems in number theory. The problem of approximating algebraic numbers is also studied as a case in the theory of transcendental numbers. Topics include the Thue-Siegel theorem, the Hermite-Lindemann theorem on the transcendency of the exponential function, and the work of C. Siegel on the transcendency of the Bessel functions and of the solutions of other differential equations. The final chapter considers the Gelfond-Schneider theorem on the transcendency of alpha to the power beta. Each proof is prefaced by a brief discussion of its scheme, which provides a helpful guide to understanding the proof's progression.

Transcendental and Algebraic Numbers (Dover Books on Mathematics)

Dover Publications

Primarily an advanced study of the modern theory of transcendental numbers, this text focuses on the theory's fundamental methods and explores its connections with other problems in number theory. Topics include the Thue-Siegel theorem; the Hermite-Lindemann theorem on the transcendency of the exponential function; the transcendency of the Bessel functions; more. 

Transcendental and Algebraic Numbers

From Pythagoras to Fermat, Euler, and latter-day thinkers, mathematicians have puzzled over the determination of integral solutions of algebraic equations with integral coefficients and with more than one unknown. This text by A. O. Gelfond, an internationally renowned leader in the study of this area, offers a relatively elementary exploration of one of the most challenging problems in number theory. Since equations in integers are encountered in issues related to physics and engineering, the solution of these equations is a matter of practical applications. Nevertheless, the theoretical interest in equations in integers is also worth pursuing because these equations are closely connected with many problems in number theory. This volume's coverage of basic theoretical aspects of such equations promises to widen the horizons of readers from advanced high school students to undergraduate majors in mathematics, physics, and engineering.

The Solution of Equations in Integers (Dover Books on Mathematics)

2014 Reprint of 1961 Edition. Full facsimile of the original edition. Not reproduced with Optical Recognition Software. The book is devoted to one of the most interesting branches of number theory, the solution of equations in integers. The solution in integers of algebraic equations in more than one unknown with integral coefficients is a most difficult problem in the theory of numbers. The theoretical importance of equations with integral coefficients is quite great as they are closely connected with many problems of number theory. Moreover, these equations are sometimes encountered in physics and so they are also important in practice. The elements of the theory of equations with integral coefficients as presented in this book are suitable for broadening the mathematical outlook of high-school students and students of pedagogical institutes. Some of the main results in the theory of the solution of equations in integers have been given and proofs of the theorems involved are supplied when they are sufficiently simple.

The Solution of Equations in Integers

Martino Fine Books