Prime numbers and computer methods for factorization

1. The Number of Primes Below a Given Limit.- What Is a Prime Number?.- The Fundamental Theorem of Arithmetic.- Which Numbers Are Primes? The Sieve of Eratosthenes.- General Remarks Concerning Computer Programs.- A Sieve Program.- Compact Prime Tables.- Hexadecimal Compact Prime Tables.- Difference Between Consecutive Primes.- The Number of Primes Below x.- Meissel's Formula.- Evaluation of Pk(x, a).- Lehmer's Formula.- Computations.- A Computation Using Meissel's Formula.- A Computation Using Lehmer's Formula.- A Computer Program Using Lehmer's Formula.- Mapes' Method.- Deduction of Formulas.- A Worked Example.- Mapes' Algorithm.- Programming Mapes' Algorithm.- Recent Developments.- Results.- Computational Complexity.- Comparison Between the Methods Discussed.- 2. The Primes Viewed at Large.- No Polynomial Can Produce Only Primes.- Formulas Yielding All Primes.- The Distribution of Primes Viewed at Large. Euclid's Theorem.- The Formulas of Gauss and Legendre for ?(x). The Prime Number Theorem.- The Chebyshev Function ?(x).- The Riemann Zeta-function.- The Zeros of the Zeta-function.- Conversion From f(x) Back to ?(x).- The Riemann Prime Number Formula.- The Sign of li x ? ?(x).- The Influence of the Complex Zeros of ?(s) on ?(x).- The Remainder Term in the Prime Number Theorem.- Effective Inequalities for ?(x), pn, and ?(x).- The Number of Primes in Arithmetic Progressions.- 3. Subtleties in the Distribution of Primes.- The Distribution of Primes in Short Intervals.- Twins and Some Other Constellations of Primes.- Admissible Constellations of Primes.- The Hardy-Littlewood Constants.- The Prime k-Tuples Conjecture.- Theoretical Evidence in Favour of the Prime k-Tuples Conjecture.- Numerical Evidence in Favour of the Prime k-Tuples Conjecture.- The Second Hardy-Littlewood Conjecture.- The Midpoint Sieve.- Modification of the Midpoint Sieve.- Construction of Superdense Admissible Constellations.- Some Dense Clusters of Primes.- The Distribution of Primes Between the Two Series 4n + 1 and 4n + 3.- Graph of the Function ?4,3(x) ? ?4,1(x).- The Negative Regions.- The Negative Blocks.- Large Gaps Between Consecutive Primes.- The Cramer Conjecture.- 4. The Recognition of Primes.- Tests of Primality and of Compositeness.- Factorization Methods as Tests of Compositeness.- Fermat's Theorem as Compositeness Test.- Fermat's Theorem as Primality Test.- Pseudoprimes and Probable Primes.- A Computer Program for Fermat's Test.- The Labor Involved in a Fermat Test.- Carmichael Numbers.- Euler Pseudoprimes.- Strong Pseudoprimes and a Primality Test.- A Computer Program for Strong Pseudoprime Tests.- Counts of Pseudoprimes and Carmichael Numbers.- Rigorous Primality Proofs.- Lehmer's Converse of Fermat's Theorem.- Formal Proof of Theorem 4.3.- Ad Hoc Search for a Primitive Root.- The Use of Several Bases.- Fermat Numbers and Pepin's Theorem.- Cofactors of Fermat Numbers.- Generalized Fermat Numbers.- A Relaxed Converse of Fermat's Theorem.- Proth's Theorem.- Tests of Compositeness for Numbers of the form N = h * 2n +- k.- An Alternative Approach.- Certificates of Primality.- Primality Tests of Lucasian Type.- Lucas Sequences.- The Fibonacci Numbers.- Large Subscripts.- An Alternative Deduction.- Divisibility Properties of the Numbers Un.- Primality Proofs by Aid of Lucas Sequences.- Lucas Tests for Mersenne Numbers.- A Relaxation of Theorem 4.8.- Pocklington's Theorem.- Lehmer-Pocklington's Theorem.- Pocklington-Type Theorems for Lucas Sequences.- Primality Tests for Integers of the form N = h * 2n ? 1, when 3?h.- Primality Tests for N = h * 2n ? 1, when 3?h.- The Combined N ? 1 and N + 1 Test.- Lucas Pseudoprimes.- Modern Primality Proofs.- The Jacobi Sum Primality Test.- Three Lemmas.- Lenstra's Theorem.- The Sets P and Q.- Running Time for the APRCL Test.- Elliptic Curve Primality Proving, ECPP.- The Goldwasser-Kilian Test.- Atkin's Test.- 5. Classical Methods of Factorization.- When Do We Attempt Factorization?.- Trial Division.- A Computer Implementation of Trial Division.- Euclid's Algorithm as an Aid to Factorization.- Fermat's Factoring Method.- Legendre's Congruence.- Euler's Factoring Method.- Gauss' Factoring Method.- Legendre's Factoring Method.- The Number of Prime Factors of Large Numbers.- How Does a Typical Factorization Look?.- The Erd?s-Kac Theorem.- The Distribution of Prime Factors of Various Sizes.- Dickman's Version of Theorem 5.4.- A More Detailed Theory.- The Size of the kth Largest Prime Factor of N.- Smooth Integers.- Searching for Factors of Certain Forms.- Legendre's Theorem for the Factors of N = an +- bn.- Adaptation to Search for Factors of the Form p = 2kn + 1.- Adaptation of Trial Division.- Adaptation of Fermat's Factoring Method.- Adaptation of Euclid's Algorithm as an Aid to Factorization.- 6. Modem Factorization Methods.- Choice of Method.- Pollard's (p ? 1)-Method.- Phase 2 of the (p ? 1)-Method.- The (p + 1)-Method.- Pollard's rho Method.- A Computer Program for Pollard's rho Method.- An Algebraic Description of Pollard's rho Method.- Brent's Modification of Pollard's rho Method.- The Pollard-Brent Method for p = 2kn + 1.- Shanks' Factoring Method SQUFOF.- A Computer Program for SQUFOF.- Comparison Between Pollard's rho Method and SQUFOF.- Morrison and Brillhart's Continued Fraction Method CFRAC.- The Factor Base.- An Example of a Factorization with CFRAC.- Further Details of CFRAC.- The Early Abort Strategy.- Results Achieved with CFRAC.- Running Time Analysis of CFRAC.- The Quadratic Sieve, QS.- Smallest Solutions to Q(x) ? 0 mod p.- Special Factors.- Results Achieved with QS.- The Multiple Polynomial Quadratic Sieve, MPQS.- Results Achieved with MPQS.- Running Time Analysis of QS and MPQS.- The Schnorr-Lenstra Method.- Two Categories of Factorization Methods.- Lenstra's Elliptic Curve Method, ECM.- Phase 2 of ECM.- The Choice of A, B, and P1.- Running Times of ECM.- Recent Results Achieved with ECM.- The Number Field Sieve, NFS.- Factoring Both in Z and in Z(z).- A Numerical Example.- The General Number Field Sieve, GNFS.- Running Times of NFS and GNFS.- Results Achieved with NFS. Factorization of F9.- Strategies in Factoring.- How Fast Can a Factorization Algorithm Be?.- 7. Prime Numbers and Cryptography.- Practical Secrecy.- Keys in Cryptography.- Arithmetical Formulation.- RSA Cryptosystems.- How to Find the Recovery Exponent.- A Worked Example.- Selecting Keys.- Finding Suitable Primes.- The Fixed Points of an RSA System.- How Safe is an RSA Cryptosystem?.- Superior Factorization.- 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 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-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.