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Elementary number theory : primes, congruences, and secrets : a computational approach / William Stein

Main Author Stein, William A. 1974- Country Estados Unidos. Publication New York : Springer, cop. 2009 Description X, 166 p. : il., gráficos ; 24 cm Series Undergraduate texts in mathematics , 0172-6056 ISBN 978-0-387-85524-0 CDU 511 511.2
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Item type Current location Call number Status Date due Barcode Item holds Course reserves
Monografia Biblioteca Geral da Universidade do Minho
BGUM 511 - S Available 395993

Licenciatura em Ciências da Computação Teoria de Números Computacional 2º semestre

Monografia Biblioteca Geral da Universidade do Minho
BGUM 511 - S Available 397628
Total holds: 0

Enhanced descriptions from Syndetics:

This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. It grew out of undergr- uate courses that the author taught at Harvard, UC San Diego, and the University of Washington. The systematic study of number theory was initiated around 300B. C. when Euclid proved that there are in?nitely many prime numbers, and also cleverly deduced the fundamental theorem of arithmetic, which asserts that every positive integer factors uniquely as a product of primes. Over a thousand years later (around 972A. D. ) Arab mathematicians formulated the congruent number problem that asks for a way to decide whether or not a given positive integer n is the area of a right triangle, all three of whose sides are rational numbers. Then another thousand years later (in 1976), Di?e and Hellman introduced the ?rst ever public-key cryptosystem, which enabled two people to communicate secretely over a public communications channel with no predetermined secret; this invention and the ones that followed it revolutionized the world of digital communication. In the 1980s and 1990s, elliptic curves revolutionized number theory, providing striking new insights into the congruent number problem, primality testing, publ- key cryptography, attacks on public-key systems, and playing a central role in Andrew Wiles' resolution of Fermat's Last Theorem.

Reviews provided by Syndetics


The cliche that number theory, ever the purest mathematics, now yields very practical applications barely tells the story. Teach undergraduate number theory today, and students demand to hear about public-key cryptography and related technologies. So though the actual mathematical content in this work (congruences, quadratic reciprocity, continued fractions) diverges not so much from R. D. Carmichael's The Theory of Numbers and Diophantine Analysis (1915), emphasis has shifted. If number theory now has applications to computation, computation returns the favor by providing methods for attacking problems that once seemed insurmountable. Moreover, the possibility of applying computation shifts theoretical priorities--many a hot open problem now speaks to the existence or behavior of an algorithm. If a century ago, number theorists wielded ad hoc methods, they now have very sophisticated machinery. Stein (Univ. of Washington) serves undergraduates well by omitting advanced tools but opening the way by intimating their power. A final chapter treats elliptic curves, a classical subject that has acquired modern centrality. Stein, remarkably, omits any mention of Fermat's Last Theorem, the holy grail of number theory until its solution in 1994. Rather, he frames the sophisticated Birch and Swinnerton-Dyer conjecture as the new canonical challenge for the future. Summing Up: Recommended. All undergraduate students, professionals, and general readers. D. V. Feldman University of New Hampshire

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