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31: 27.19 Methods of Computation: Factorization
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►Techniques for factorization of integers fall into three general classes: Deterministic algorithms, Type I probabilistic algorithms whose expected running time depends on the size of the smallest prime factor, and Type II probabilistic algorithms whose expected running time depends on the size of the number to be factored.
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►As of January 2009 the largest prime factors found by these methods are a 19-digit prime for Brent–Pollard rho, a 58-digit prime for Pollard , and a 67-digit prime for ecm.
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►These algorithms include the Continued Fraction Algorithm (cfrac), the Multiple Polynomial Quadratic Sieve (mpqs), the General
Number Field Sieve (gnfs), and the Special Number Field Sieve (snfs).
…The snfs can be applied only to numbers that are very close to a power of a very small base.
The largest composite numbers that have been factored by other Type II probabilistic algorithms are a 63-digit integer by cfrac, a 135-digit integer by mpqs, and a 182-digit integer by gnfs.
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32: 26.17 The Twelvefold Way
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►The twelvefold way gives the number of mappings from set of objects to set of objects (putting balls from set into boxes in set ).
…In this table is Pochhammer’s symbol, and and are defined in §§26.8(i) and 26.9(i).
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33: 24.20 Tables
§24.20 Tables
… ►Wagstaff (1978) gives complete prime factorizations of and for and , respectively. …34: 24.9 Inequalities
35: 27.16 Cryptography
§27.16 Cryptography
… ►The primes are kept secret but their product , an 800-digit number, is made public. …With the most efficient computer techniques devised to date (2010), factoring an 800-digit number may require billions of years on a single computer. For this reason, the codes are considered unbreakable, at least with the current state of knowledge on factoring large numbers. … ►Thus, and . …36: 27.21 Tables
§27.21 Tables
… ►Bressoud and Wagon (2000, pp. 103–104) supplies tables and graphs that compare , and . … ►Lehmer (1941) gives a comprehensive account of tables in the theory of numbers, including virtually every table published from 1918 to 1941. … ►No sequel to Lehmer (1941) exists to date, but many tables of functions of number theory are included in Unpublished Mathematical Tables (1944).37: 4.19 Maclaurin Series and Laurent Series
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►In (4.19.3)–(4.19.9), are the Bernoulli numbers and are the Euler numbers (§§24.2(i)–24.2(ii)).
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4.19.3
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4.19.4
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4.19.5
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4.19.6
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38: 24.4 Basic Properties
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§24.4(iv) Finite Expansions
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24.4.15
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24.4.16
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