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OverviewFull Product DetailsAuthor: Hans RieselPublisher: Springer-Verlag New York Inc. Imprint: Springer-Verlag New York Inc. Edition: Softcover reprint of the original 2nd ed. 1994 Volume: 126 Dimensions: Width: 15.50cm , Height: 2.40cm , Length: 23.50cm Weight: 0.735kg ISBN: 9781461266815ISBN 10: 1461266815 Pages: 464 Publication Date: 30 September 2012 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: Manufactured on demand  We will order this item for you from a manufactured on demand supplier. Table of Contents1. The Number of Primes Below a Given Limit.- 2. The Primes Viewed at Large.- 3. Subtleties in the Distribution of Primes.- 4. The Recognition of Primes.- 5. Classical Methods of Factorization.- 6. Modem Factorization Methods.- 7. Prime Numbers and Cryptography.- Appendix 1. Basic Concepts in Higher Algebra.- Modules.- Euclid’s Algorithm.- The Labor Involved in Euclid’s Algorithm.- A Definition Taken from the Theory of Algorithms.- A Computer Program for Euclid’s Algorithm.- Reducing the Labor.- Binary Form of Euclid’s Algorithm.- Groups.- Lagrange’s Theorem. Cosets.- Abstract Groups. Isomorphic Groups.- The Direct Product of Two Given Groups.- Cyclic Groups.- Rings.- Zero Divisors.- Fields.- Mappings. Isomorphisms and Homomorphisms.- Group Characters.- The Conjugate or Inverse Character.- Homomorphisms and Group Characters.- Appendix 2. Basic Concepts in Higher Arithmetic.- Divisors. Common Divisors.- The Fundamental Theorem of Arithmetic.- Congruences.- Linear Congruences.- Linear Congruences and Euclid’s Algorithm.- Systems of Linear Congruences.- Carmichael’s Function.- Carmichael’s Theorem.- Appendix 3. Quadratic Residues.- Legendre’s Symbol.- Arithmetic Rules for Residues and Non-Residues.- The Law of Quadratic Reciprocity.- Jacobi’s Symbol.- Appendix 4. The Arithmetic of Quadratic Fields.- Appendix 5. Higher Algebraic Number Fields.- Algebraic Numbers.- Appendix 6. Algebraic Factors.- Factorization of Polynomials.- The Cyclotomic Polynomials.- Aurifeuillian Factorizations.- Factorization Formulas.- The Algebraic Structure of Aurifeuillian Numbers.- Appendix 7. Elliptic Curves.- Cubics.- Rational Points on Rational Cubics.- Homogeneous Coordinates.- Elliptic Curves.- Rational Points on Elliptic Curves.- Appendix 8. Continued Fractions.- What Isa Continued Fraction?.- Regular Continued Fractions. Expansions.- Evaluating a Continued Fraction.- Continued Fractions as Approximations.- Euclid’s Algorithm and Continued Fractions.- Linear Diophantine Equations and Continued Fractions.- A Computer Program.- Continued Fraction Expansions of Square Roots.- Proof of Periodicity.- The Maximal Period-Length.- Short Periods.- Continued Fractions and Quadratic Residues.- Appendix 9. Multiple-Precision Arithmetic.- Various Objectives for a Multiple-Precision Package.- How to Store Multi-Precise Integers.- Addition and Subtraction of Multi-Precise Integers.- Reduction in Length of Multi-Precise Integers.- Multiplication of Multi-Precise Integers.- Division of Multi-Precise Integers.- Input and Output of Multi-Precise Integers.- A Complete Package for Multiple-Precision Arithmetic.- A Computer Program for Pollard’s rho Method.- Appendix 10. Fast Multiplication of Large Integers.- The Ordinary Multiplication Algorithm.- Double Length Multiplication.- Recursive Use of Double Length Multiplication Formula.- A Recursive Procedure for Squaring Large Integers.- Fractal Structure of Recursive Squaring.- Large Mersenne Primes.- Appendix 11. The Stieltjes Integral.- Functions With Jump Discontinuities.- The Riemann Integral.- Definition of the Stieltjes Integral.- Rules of Integration for Stieltjes Integrals.- Integration by Parts of Stieltjes Integrals.- The Mean Value Theorem.- Applications.- Tables. For Contents.- List of Textbooks.ReviewsHere is an outstanding technical monograph on recursive number theory and its numerous automated techniques. It successfully passes a critical milestone not allowed to many books, viz., a second edition. Many good things have happened to computational number theory during the ten years since the first edition appeared and the author includes their highlights in great depth. Several major sections have been rewritten and totally new sections have been added. The new material includes advances on applications of the elliptic curve method, uses of the number field sieve, and two new appendices on the basics of higher algebraic number fields and elliptic curves. Further, the table of prime factors of Fermat numbers has been significantly up-dated. ...Several other tables have been added so as to provide data to look for large prime factors of certain 'generalized' Fermat numbers, while several other tables on special numbers were simply deleted in the second edition. Still one can make several perplexing assertions or challenges: (1) prove that F\sb 5, F\sb 6, F\sb 7, F\sb 8 are the only four consecutive Fermat numbers which are bi-composite; (2) Show that F\sb{14} is bi- composite. (This accounts for the difficulty in finding a prime factor for it.) (3) What is the smallest Fermat quadri-composite?; and (4) Does there exist a Fermat number with an arbitrarily prescribed number of prime factors? All in all, this handy volume continues to be an attractive combination of number-theoretic precision, practicality, and theory with a rich blend of computer science. -Zentralblatt Math Here is an outstanding technical monograph on recursive number theory and its numerous automated techniques. It successfully passes a critical milestone not allowed to many books, viz., a second edition. Many good things have happened to computational number theory during the ten years since the first edition appeared and the author includes their highlights in great depth. Several major sections have been rewritten and totally new sections have been added. The new material includes advances on applications of the elliptic curve method, uses of the number field sieve, and two new appendices on the basics of higher algebraic number fields and elliptic curves. Further, the table of prime factors of Fermat numbers has been significantly up-dated. ...Several other tables have been added so as to provide data to look for large prime factors of certain 'generalized' Fermat numbers, while several other tables on special numbers were simply deleted in the second edition. Still one can make several perplexing assertions or challenges: (1) prove that F\sb 5, F\sb 6, F\sb 7, F\sb 8 are the only four consecutive Fermat numbers which are bi-composite; (2) Show that F\sb{14} is bi- composite. (This accounts for the difficulty in finding a prime factor for it.) (3) What is the smallest Fermat quadri-composite?; and (4) Does there exist a Fermat number with an arbitrarily prescribed number of prime factors? All in all, this handy volume continues to be an attractive combination of number-theoretic precision, practicality, and theory with a rich blend of computer science. -Zentralblatt Math Author InformationTab Content 6Author Website:Countries AvailableAll regions |
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