Prime Numbers and Computer Methods for FactorizationSpringer Science & Business Media, 1994 - Всего страниц: 464 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 Cramér 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?.- Tri |
Содержание
The Number of Primes Below a Given Limit | 1 |
The Primes Viewed at Large | 37 |
Subtleties in the Distribution of Primes | 60 |
The Recognition of Primes | 84 |
Classical Methods of Factorization | 141 |
Modern Factorization Methods | 173 |
Prime Numbers and Cryptography | 226 |
Basic Concepts in Higher Algebra | 239 |
Higher Algebraic Number Fields | 297 |
Algebraic Factors | 304 |
Elliptic Curves | 317 |
Continued Fractions | 327 |
MultiplePrecision Arithmetic | 343 |
Fast Multiplication of Large Integers | 357 |
The Stieltjes Integral | 365 |
Tables For Contents see page | 374 |
Basic Concepts in Higher Arithmetic | 261 |
Quadratic Residues | 276 |
The Arithmetic of Quadratic Fields | 286 |
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Часто встречающиеся слова и выражения
addition algebraic algorithm Appendix applying approximately arithmetic base BEGIN calculation called chosen coefficients combination Comp completely composite congruence consider constellation construct contains continued fraction corresponding curve defined depends described digits divide division divisors elements equation Euclid's algorithm example exist expansion expression fact Fermat field finally formula function give given integers interval known largest leading length limit Lucas Math means method mod N multiplication obtain occur operations performed period polynomial positive possible precisely primality tests prime factors primitive problem procedure proof proved pseudoprime quadratic residues rational reader reason reduced residue classes result root rule running satisfying sequence sieve simple solution square steps stored Suppose Table Theorem theory trial unit values written zero