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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.
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.
This is a tutorial explaining how to use the free and open source
mathematical software package Sage (version 6.1.1). Sage and this
can be downloaded free from the website: http: //www.sagemath.org/.
Copyright: (c) 2014 Creative Commons Attribution-ShareAlike 3.0.
Royalties go directly to the Sage Foundation.
Prime numbers are beautiful, mysterious, and beguiling mathematical
objects. The mathematician Bernhard Riemann made a celebrated
conjecture about primes in 1859, the so-called Riemann hypothesis,
which remains one of the most important unsolved problems in
mathematics. Through the deep insights of the authors, this book
introduces primes and explains the Riemann hypothesis. Students
with a minimal mathematical background and scholars alike will
enjoy this comprehensive discussion of primes. The first part of
the book will inspire the curiosity of a general reader with an
accessible explanation of the key ideas. The exposition of these
ideas is generously illuminated by computational graphics that
exhibit the key concepts and phenomena in enticing detail. Readers
with more mathematical experience will then go deeper into the
structure of primes and see how the Riemann hypothesis relates to
Fourier analysis using the vocabulary of spectra. Readers with a
strong mathematical background will be able to connect these ideas
to historical formulations of the Riemann hypothesis.
Prime numbers are beautiful, mysterious, and beguiling mathematical
objects. The mathematician Bernhard Riemann made a celebrated
conjecture about primes in 1859, the so-called Riemann hypothesis,
which remains one of the most important unsolved problems in
mathematics. Through the deep insights of the authors, this book
introduces primes and explains the Riemann hypothesis. Students
with a minimal mathematical background and scholars alike will
enjoy this comprehensive discussion of primes. The first part of
the book will inspire the curiosity of a general reader with an
accessible explanation of the key ideas. The exposition of these
ideas is generously illuminated by computational graphics that
exhibit the key concepts and phenomena in enticing detail. Readers
with more mathematical experience will then go deeper into the
structure of primes and see how the Riemann hypothesis relates to
Fourier analysis using the vocabulary of spectra. Readers with a
strong mathematical background will be able to connect these ideas
to historical formulations of the Riemann hypothesis.
This marvellous and highly original book fills a significant gap in
the extensive literature on classical modular forms. This is not
just yet another introductory text to this theory, though it could
certainly be used as such in conjunction with more traditional
treatments. Its novelty lies in its computational emphasis
throughout: Stein not only defines what modular forms are, but
shows in illuminating detail how one can compute everything about
them in practice. This is illustrated throughout the book with
examples from his own (entirely free) software package SAGE, which
really bring the subject to life while not detracting in any way
from its theoretical beauty. The author is the leading expert in
computations with modular forms, and what he says on this subject
is all tried and tested and based on his extensive experience. As
well as being an invaluable companion to those learning the theory
in a more traditional way, this book will be a great help to those
who wish to use modular forms in appl --John E. Cremona, University
of Nottingham William Stein is an associate professor of
mathematics at the University of Washington at Seattle. He earned a
PhD in mathematics from UC Berkeley and has held positions at
Harvard University and UC San Diego. His current research interests
lie in modular forms, elliptic curves, and computational
mathematics.
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