03753nam a22003855i 4500001001800000003000900018005001700027007001500044008004100059020001800100020001900118024002400137100002900161245013500190264004600325300004100371336002600412337002600438338003600464347002400500490005100524505014600575520229400721650001703015650002503032650001903057650001703076650001903093650002403112710003403136773002003170776003603190830005103226856009003277978-0-387-85525-7DE-He21320260521092013.0cr nn 008mamaa110406s2009    xxu|    s    |||| 0|eng d  a9780387855257  a997803878552577 a10.1007/b132792doi1 aStein, William.eauthor.10aElementary Number Theory: Primes, Congruences, and Secretsh[electronic resource] :bA Computational Approach /cby William Stein. 1aNew York, NY :bSpringer New York,c2009.  aX, 170p. 45 illus.bonline resource.  atextbtxt2rdacontent  acomputerbc2rdamedia  aonline resourcebcr2rdacarrier  atext filebPDF2rda1 aUndergraduate Texts in Mathematics,x0172-60560 aPrime Numbers -- The Ring of Integers Modulo n -- Public-key Cryptography -- Quadratic Reciprocity -- Continued Fractions -- Elliptic Curves.  aThe systematic study of number theory was initiated around 300B.C. when Euclid proved that there are infinitely many prime numbers. At the same time, he also cleverly deduced the fundamental theorem of arithmetic, which asserts that every positive integer factors uniquely as a product of primes. Over 1000 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 1000 years later (in 1976), Diffie and Hellman introduced the first ever public-key cryptosystem, which enabled two people to communicate secretly 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, public-key cryptography, attacks on public-key systems, and playing a central role in Andrew Wiles' resolution of Fermat's Last Theorem. Today, pure and applied number theory is an exciting mix of simultaneously broad and deep theory, which is constantly informed and motivated by algorithms and explicit computation. Active research is underway that promises to resolve the congruent number problem, deepen our understanding into the structure of prime numbers, and both challenge and improve our ability to communicate securely. The goal of this book is to bring the reader closer to this world. Each chapter contains exercises, and throughout the text there are examples of calculations done using the powerful free open source mathematical software system Sage. The reader should know how to read and write mathematical proofs and must know the basics of groups, rings, and fields. Thus, the prerequisites for this book are more than the prerequisites for most elementary number theory books, while still being aimed at undergraduates. William Stein is an Associate Professor of Mathematics at the University of Washington. He is also the author of Modular Forms, A Computational Approach (AMS 2007), and the lead developer of the open source software, Sage. 0aMATHEMATICS. 0aGEOMETRY, ALGEBRAIC. 0aNUMBER THEORY.14aMATHEMATICS.24aNUMBER THEORY.24aALGEBRAIC GEOMETRY.2 aSpringerLink (Online service)0 tSpringer eBooks08iPrinted edition:z9780387855240 0aUndergraduate Texts in Mathematics,x0172-605640uhttp://dx.doi.org/10.1007/b13279zVer el texto completo en las instalaciones del CICY