Prime Numbers and Computer Methods for FactorizationSpringer Science & Business Media, 14 mars 2013 - 464 pages In this book the author treats four fundamental and apparently simple problems. They are: the number of primes below a given limit, the ap proximate number of primes, the recognition of prime numbers and the factorization of large numbers. A chapter on the details of the distribution of the primes is included as well as a short description of a recent applica tion of prime numbers, the so-called RSA public-key cryptosystem. The author is also giving explicit algorithms and computer programs. Whilst not claiming completeness, the author has tried to give all important results known, including the latest discoveries. The use of computers has in this area promoted a development which has enormously enlarged the wealth of results known and that has made many older works and tables obsolete. As is often the case in number theory, the problems posed are easy to understand but the solutions are theoretically advanced. Since this text is aimed at the mathematically inclined layman, as well as at the more advanced student, not all of the proofs of the results given in this book are shown. Bibliographical references in these cases serve those readers who wish to probe deeper. References to recent original works are also given for those who wish to pursue some topic further. Since number theory is seldom taught in basic mathematics courses, the author has appended six sections containing all the algebra and number theory required for the main body of the book. |
Table des matières
1 | |
8 | |
Evaluation of Px | 18 |
A Computer Program Using Lehmers Formula | 24 |
Recent Developments | 36 |
Subtleties in the Distribution of Primes | 64 |
The Prime kTuples Conjecture | 70 |
Construction of Superdense Admissible Constellations | 76 |
91 | 241 |
Basic Concepts in Higher Arithmetic | 262 |
92 | 263 |
Quadratic Residues | 278 |
The Arithmetic of Quadratic Fields | 289 |
Continued Fractions | 300 |
Algebraic Factors | 318 |
MultiplePrecision Arithmetic | 332 |
The Negative Blocks | 83 |
The Recognition of Primes | 90 |
47 | 114 |
Factorization | 146 |
0000 | 204 |
Prime Numbers and Cryptography | 223 |
90 | 228 |
Basic Concepts in Higher Algebra | 237 |
95 | 337 |
Fast Multiplication of Large Integers | 348 |
The Stieltjes Integral | 358 |
Tables | 368 |
96 | 378 |
100 | 385 |
460 | |
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Expressions et termes fréquents
addsub admissible constellation algebraic factors Appendix approximately arithmetic b₁ BEGIN binary calculation Carmichael numbers coefficients Comp composite number computer program conjecture continued fraction expansion cyclic group cyclotomic D. H. Lehmer digits divisors elements Euclid's algorithm Euler's criterion example factor base factorization methods Fermat Numbers Fermat's theorem formula function Gauss give GOTO interval large numbers Legendre's Lehmer's ln ln Lucas sequence Math module multiple-precision multiplication number of prime number theory odd prime p-method p₁ Pollard's method polynomial possible precisely primality tests prime factors prime k-tuples conjecture Prime Number Theorem prime table primitive residue classes probable prime procedure proof proved quadratic non-residue quadratic residues quotient reader recursion regular continued fraction residue classes residue classes mod result Riemann search limit Shanks sieve of Eratosthenes small prime factors solutions trial division vector zero