On 2008-07-05 Man Kam Tam, Calexico, CA USA wrote: Harold M. Edwards´ ´Higher Arithmetic: An Algorithmic Introduction to Number Theory´ has 31 chapters, but merely 210 pages. So each chapter has not only six pages but also a very specific theme. The book would interest the readers whom would like to learn more about number theory, Diophantine equation, and Hilbert´s tenth problem.
The book begins with a few interesting comments such as ´I ... prefer arithmetic to number theory ... Students enjoy and ... profit from doing computational assignments ... The basic questions of number theory can be stated in terms of congruence ...´
As a matter of fact, programming the provided algorithms makes learning number theory interesting. The interested readers may consider the programming language Octave (Matlab) or Maxima (Maple), because Matlab and Maple are both popular mathematical software. On the other hand, doing the exercises is truly profitable as well, since the provided solutions are thorough, except on utilizing the Chinese Remainder Theorem on searching the square root of A mod B.
A feature of the book is that it has a theme. The theme is to find all the solutions x and y such that Ax^2 + B = y^2, where A, B, x, and y are integers. In order to search for the solutions x and y, Edwards provides an algorithm named comparison algorithm. By inputting the square root of A mod B, A, and B recursively until F is equal to 1, the desired solution can be obtained. Other interesting features include primality testing and the RSA cipher system. The mechanism of the RSA cipher system is explained clearly; the best I ever read. The book also provides a primality testing algorithm to determine ´whether a number is composite, but it can never prove that a number is prime.´. And summed up by saying A Book Which Make Learning Number Theory Interesting. Currently Higher Arithmetic (Student Mathematical Library) has an overall rating of 10 over 10.
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American Mathematical Society claimed Although number theorists have sometimes shunned and even disparaged computation in the past, today´s applications of number theory to cryptography and computer security demand vast arithmetical computations. These demands have shifted the focus of studies in number theory and have changed attitudes toward computation itself. The important new applications have attracted a great many students to number theory, but the best reason for studying the subject remains what it was when Gauss published his classic Disquisitiones Arithmeticae in 1801: Number theory is the equal of Euclidean geometry--some would say it is superior to Euclidean geometry--as a model of pure, logical, deductive thinking. An arithmetical computation, after all, is the purest form of deductive argument. Higher Arithmetic explains number theory in a way that gives deductive reasoning, including algorithms and computations, the central role. Hands-on experience with the application of algorithms to computational examples enables students to master the fundamental ideas of basic number theory. This is a worthwhile goal for any student of mathematics and an essential one for students interested in the modern applications of number theory. Harold M. Edwards is Emeritus Professor of Mathematics at New York University. His previous books are Advanced Calculus (1969, 1980, 1993), Riemann´s Zeta Function (1974, 2001), Fermat´s Last Theorem (1977), Galois Theory (1984), Divisor Theory (1990), Linear Algebra (1995), and Essays in Constructive Mathematics (2005). For his masterly mathematical exposition he was awarded a Steele Prize as well as a Whiteman Prize by the American Mathematical Society.
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