Prime Numbers and Computer Methods for Factorization - Brossura

Riesel, H.

Brossura

ISBN 10: 1475710909 ISBN 13: 9781475710908

Casa editrice: Birkhäuser, 2012

Riassunto

1. 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. Factorization.- 6. Prime Numbers and Cryptography.- Appendix 1. Basic Concepts in Higher Algebra.- Modules.- Euclid's Algorithm.- The Labour Involved in Euclid's Algorithm.- A Definition Taken from the Theory of Algorithms.- A Computer Program for Euclid's Algorithm.- Reducing the Labour.- 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. Continued Fractions.- What Is a 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-Lenath.- Short Periods.- Continued Fractions and Quadratic Residues.- Appendix 6. Algebraic Factors.- Factorization of Polynomials.- The Cyclotomic Polynomials.- Aurifeuillian Factorizations.- Factorization Formulas.- The Algebraic Structure of Aurifeuillian Numbers.- Appendix 7. 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.- Appendix 8. 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 9. 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.- Tatle 1. The Primes Belnw 12553.- Table 4. Factors of Fermat Numbers.- Table 7. Factors of Mersenne Numbers.- Table 32. Quadratic Residues.- Table 33. Gauss' formulas for Cyclotomic Polynomials.- Table 34. Lucas' formulas for Cyclotomic Polynomials.

Dettagli dell?edizione

• Editore: Birkhäuser
• Data di pubblicazione: 2012
• ISBN 10: 1475710909
• ISBN 13: 9781475710908
• Rilegatura: Paperback
• Lingua: Inglese
• Contatto del produttore: non disponibile