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Factorization and Primality Testing

Author: David M Bressoud
Publisher: New York, NY : Springer New York, 1989.
Series: Undergraduate texts in mathematics.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
"About binomial theorems I'm teeming with a lot of news, With many cheerful facts about the square on the hypotenuse."--William S. Gilbert (The Pirates of Penzance, Act I) The question of divisibility is arguably the oldest problem in mathematics. Ancient peoples observed the cycles of nature: the day, the lunar month, and the year, and assumed that each divided evenly into the next. Civilizations as separate as the  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: David M Bressoud
ISBN: 9781461245445 1461245443
OCLC Number: 852791497
Description: 1 online resource (260 pages 2 illustrations).
Contents: 1 Unique Factorization and the Euclidean Algorithm --
1.1 A theorem of Euclid and some of its consequences --
1.2 The Fundamental Theorem of Arithmetic --
1.3 The Euclidean Algorithm --
1.4 The Euclidean Algorithm in practice --
1.5 Continued fractions, a first glance --
1.6 Exercises --
2 Primes and Perfect Numbers --
2.1 The Number of Primes --
2.2 The Sieve of Eratosthenes --
2.3 Trial Division --
2.4 Perfect Numbers --
2.5 Mersenne Primes --
2.6 Exercises --
3 Fermat, Euler, and Pseudoprimes --
3.1 Fermat's Observation --
3.2 Pseudoprimes --
3.3 Fast Exponentiation --
3.4 A Theorem of Euler --
3.5 Proof of Fermat's Observation --
3.6 Implications for Perfect Numbers --
3.7 Exercises --
4 The RSA Public Key Crypto-System --
4.1 The Basic Idea --
4.2 An Example --
4.3 The Chinese Remainder Theorem --
4.4 What if the Moduli are not Relatively Prime? --
4.5 Properties of Euler's ø Function --
Exercises --
5 Factorization Techniques from Fermat to Today --
5.1 Fermat's Algorithm --
5.2 Kraitchik's Improvement --
5.3 Pollard Rho --
5.4 Pollard p --
1 --
5.5 Some Musings --
5.6 Exercises --
6 Strong Pseudoprimes and Quadratic Residues --
6.1 The Strong Pseudoprime Test --
6.2 Refining Fermat's Observation --
6.3 No 'Strong' Carmichael Numbers --
6.4 Exercises --
7 Quadratic Reciprocity --
7.1 The Legendre Symbol --
7.2 The Legendre symbol for small bases --
7.3 Quadratic Reciprocity --
7.4 The Jacobi Symbol --
7.5 Computing the Legendre Symbol --
7.6 Exercises --
8 The Quadratic Sieve --
8.1 Dixon's Algorithm --
8.2 Pomerance's Improvement --
8.3 Solving Quadratic Congruences --
8.4 Sieving --
8.5 Gaussian Elimination --
8.6 Large Primes and Multiple Polynomials --
8.7 Exercises --
9 Primitive Roots and a Test for Primality --
9.1 Orders and Primitive Roots --
9.2 Properties of Primitive Roots --
9.3 Primitive Roots for Prime Moduli --
9.4 A Test for Primality --
9.5 More on Primality Testing --
9.6 The Rest of Gauss' Theorem --
9.7 Exercises --
10 Continued Fractions --
10.1 Approximating the Square Root of 2 --
10.2 The Bháscara-Brouncker Algorithm --
10.3 The Bháscara-Brouncker Algorithm Explained --
10.4 Solutions Really Exist --
10.5 Exercises --
11 Continued Fractions Continued, Applications --
11.1 CFRAC --
11.2 Some Observations on the Bháscara-Brouncker Algorithm --
11.3 Proofs of the Observations --
11.4 Primality Testing with Continued Fractions --
11.5 The Lucas-Lehmer Algorithm Explained --
11.6 Exercises --
12 Lucas Sequences --
12.1 Basic Definitions --
12.2 Divisibility Properties --
12.3 Lucas' Primality Test --
12.4 Computing the V's --
12.5 Exercises --
13 Groups and Elliptic Curves --
13.1 Groups --
13.2 A General Approach to Primality Tests --
13.3 A General Approach to Factorization --
13.4 Elliptic Curves --
13.5 Elliptic Curves Modulo p --
13.6 Exercises --
14 Applications of Elliptic Curves --
14.1 Computation on Elliptic Curves --
14.2 Factorization with Elliptic Curves --
14.3 Primality Testing --
14.4 Quadratic Forms --
14.5 The Power Residue Symbol --
14.6 Exercises --
The Primes Below 5000.
Series Title: Undergraduate texts in mathematics.
Responsibility: by David M. Bressoud.

Abstract:

Civilizations as separate as the Egyptians of ten thousand years ago and the Central American Mayans adopted a month of thirty days and a year of twelve months. At the other extreme are those  Read more...

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