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Course in Computational Number Theory [Kõva köide]

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(Macalester College),
  • Formaat: Hardback, 384 pages, kõrgus x laius x paksus: 239x185x25 mm, kaal: 874 g
  • Ilmumisaeg: 06-Nov-2008
  • Kirjastus: John Wiley & Sons Inc
  • ISBN-10: 0470412151
  • ISBN-13: 9780470412152
Teised raamatud teemal:
Course in Computational Number TheorySuurem pilt
  • Formaat: Hardback, 384 pages, kõrgus x laius x paksus: 239x185x25 mm, kaal: 874 g
  • Ilmumisaeg: 06-Nov-2008
  • Kirjastus: John Wiley & Sons Inc
  • ISBN-10: 0470412151
  • ISBN-13: 9780470412152
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780470412152.html
Märksõnad:
A Course in Computational Number Theory uses the computer as a tool for motivation and explanation. The book is designed for the reader to quickly access a computer and begin doing personal experiments with the patterns of the integers. It presents and explains many of the fastest algorithms for working with integers. Traditional topics are covered, but the text also explores factoring algorithms, primality testing, the RSA public-key cryptosystem, and unusual applications such as check digit schemes and a computation of the energy that holds a salt crystal together. Advanced topics include continued fractions, Pell's equation, and the Gaussian primes.

The CD-ROM contains a Mathematica? package that has hundreds of functions that show step-by-step operation of famous algorithms. (The user must have Mathematica in order to use this package.) Also included is an auxiliary package that contains a database of all 53,000 integers below 10^16 that are 2- and 3-strong pseudoprimes. Users will also have access to an online guide that gives illustrative examples of each function.
Preface v
Notation xi
Fundamentals
1(40)
Introduction
1(1)
A Famous Sequence of Numbers
2(4)
The Euclidean Algorithm
6(19)
The Oldest Algorithm
Reversing the Euclidean Algorithm
The Extended GCD Algorithm
The Fundamental Theorem of Arithmetic
Two Applications
Modular Arithmetic
25(5)
Fast Powers
30(11)
A Fast Algorithm for Exponentiation
Powers of Matrices, Big-O Notation
Congruences, Equations, and Powers
41(24)
Introduction
41(1)
Solving Linear Congruences
41(8)
Linear Diophantine Equations in Two Variables
Linear Equations in Several Variables
Linear Congruences
The Conductor
An Important Quadratic Congruence
The Chinese Remainder Theorem
49(6)
PowerMod Patterns
55(4)
Fermat's Little Theorem
More Patterns in Powers
Pseudoprimes
59(6)
Using the Pseudoprime Test
Euler's Function
65(34)
Introduction
65(1)
Euler's Function
65(7)
Perfect Numbers and Their Relatives
72(9)
The Sum of Divisors Function
Perfect Numbers
Amicable, Abundant, and Deficient Numbers
Euler's Theorem
81(3)
Primitive Roots for Primes
84(6)
The Order of an Integer
Primes Have Primitive Roots
Repeating Decimals
Primitive Roots for Composites
90(3)
The Universal Exponent
93(6)
Universal Exponents
Power Towers
The Form of Carmichael Numbers
Prime Numbers
99(46)
Introduction
99(1)
The Number of Primes
100(14)
We'll Never Run Out of Primes
The Sieve of Eratosthenes
Chebyshev's Theorem and Bertrand's Postulate
Prime Testing and Certification
114(17)
Strong Pseudoprimes
Industrial-Grade Primes
Prime Certification Via Primitive Roots
An Improvement
Pratt Certificates
Refinements and Other Directions
131(10)
Other Primality Tests
Strong Liars Are Scarce
Finding the nth Prime
A Dozen Prime Mysteries
141(4)
Some Applications
145(34)
Introduction
145(1)
Coding Secrets
145(10)
Tossing a Coin into a Well
The RSA Cryptosystem
Digital Signatures
The Yao Millionaire Problem
155(3)
Check Digits
158(9)
Basic Check Digit Schemes
A Perfect Check Digit Method
Beyond Perfection: Correcting Errors
Factoring Algorithms
167(12)
Trial Division
Fermat's Algorithm
Pollard Rho
Pollard p- 1
The Current Scene
Quadratic Residues
179(22)
Introduction
179(1)
Pepin's Test
179(6)
Quadratic Residues
Pepin's Test
Primes Congruent to 1 (Mod 4)
Proof of Quadratic Reciprocity
185(9)
Gauss's Lemma
Proof of Quadratic Reciprocity
Jacobi's Extension
An Application to Factoring
Quadratic Equations
194(7)
Continued Fractions
201(46)
Introduction
201(1)
Finite Continued Fractions
202(5)
Infinite Continued Fractions
207(6)
Periodic Continued Fractions
213(14)
Pell's Equation
227(5)
Archimedes and the Sun God's Cattle
232(6)
Wurm's Version: Using Rectangular Bulls
The Real Cattle Problem
Factoring via Continued Fractions
238(9)
Prime Testing with Lucas Sequences
247(32)
Introduction
247(1)
Divisibility Properties of Lucas Sequences
248(11)
Prime Tests Using Lucas Sequences
259(20)
Lucas Certification
The Lucas--Lehmer Algorithm Explained
Lucas Pseudoprimes
Strong Quadratic Pseudoprimes
Primality Testing's Holy Grail
Prime Imaginaries and Imaginary Primes
279(54)
Introduction
279(1)
Sums of Two Squares
279(23)
Primes
The General Problem
How Many Ways
Number Theory and Salt
The Gaussian Integers
302(23)
Complex Number Theory
Gaussian Primes
The Moat Problem
The Gaussian Zoo
Higher Reciprocity
325(8)
Appendix A. Mathematics Basics
333(18)
Introduction
333(2)
Plotting
335(3)
Typesetting
338(3)
Sending Files By E-Mail
Types of Functions
341(2)
Lists
343(2)
Programs
345(2)
Solving Equations
347(2)
Symbolic Algebra
349(2)
Appendix B. Lucas Certificates Exist
351(4)
References 355(4)
Index of Mathematics Objects 359(4)
Subject Index 363
David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College and president-elect of the Mathematical Association of America. He served in the Peace Corps, teaching math and science at the Clare Hall School in Antigua, West Indies before studying with Emil Grosswald at Temple University and then teaching at Penn State for 17 years, eight of them as full professor. He chaired the Department of Mathematics and Computer Science at Macalester from 1995 until 2001. He has held visiting positions at the Institute for Advanced Study, the University of Wisconsin-Madison, the University of Minnesota, Université Louis Pasteur (Strasbourg, France), and the State College Area High School. David has received the MAA Distinguished Teaching Award (Allegheny Mountain Section), the MAA Beckenbach Book Award for Proofs and Confirmations, and has been a Pólya Lecturer for the MAA. He is a recipient of Macalester's Jefferson Award. He has published over fifty research articles in number theory, combinatorics, and special functions. His other books include Factorization and Primality Testing, Second Year Calculus from Celestial Mechanics to Special Relativity, A Radical Approach to Real Analysis (now in 2nd edition), and, with Stan Wagon, A Course in Computational Number Theory. His latest book, A Radical Approach to Lebesgue's Theory of Integration, is due out by the end of 2007. David chairs the MAA special interest group, Teaching Advanced High School Mathematics. He has chaired the AP Calculus Development Committee and has served as Director of the FIPSE-sponsored program Quantitative Methods for Public Policy. He has been involved in the activities and programs of both the Mathematical Association of America and the American Mathematical Society.