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21: 24.5 Recurrence Relations
§24.5 Recurrence Relations
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24.5.3
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24.5.5
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§24.5(ii) Other Identities
… ►§24.5(iii) Inversion Formulas
…22: 24.6 Explicit Formulas
23: 27.13 Functions
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§27.13(i) Introduction
►Whereas multiplicative number theory is concerned with functions arising from prime factorization, additive number theory treats functions related to addition of integers. …The subsections that follow describe problems from additive number theory. … ►§27.13(ii) Goldbach Conjecture
… ►§27.13(iii) Waring’s Problem
…24: 26.13 Permutations: Cycle Notation
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►The Stirling cycle numbers of the first kind, denoted by , count the number of permutations of with exactly cycles.
They are related to Stirling numbers of the first kind by
…See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations.
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►The derangement number, , is the number of elements of with no fixed points:
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►A permutation is even or odd according to the parity of the number of transpositions.
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25: Bibliography K
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Poly-Bernoulli numbers.
J. Théor. Nombres Bordeaux 9 (1), pp. 221–228.
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Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity.
J. Comput. Appl. Math. 205 (1), pp. 186–206.
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On formulas involving both the Bernoulli and Fibonacci numbers.
Scripta Math. 23, pp. 27–35.
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Remark on -adic -Bernoulli numbers.
Adv. Stud. Contemp. Math. (Pusan) 1, pp. 127–136.
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Orthogonal polynomials and computer algebra.
In Recent developments in complex analysis and computer algebra
(Newark, DE, 1997), R. P. Gilbert, J. Kajiwara, and Y. S. Xu (Eds.),
Int. Soc. Anal. Appl. Comput., Vol. 4, Dordrecht, pp. 205–234.
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26: 27.9 Quadratic Characters
§27.9 Quadratic Characters
►For an odd prime , the Legendre symbol is defined as follows. … ►
27.9.2
►If are distinct odd primes, then the quadratic reciprocity law states that
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►If an odd integer has prime factorization , then the Jacobi symbol
is defined by , with .
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27: Bibliography T
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New congruences for the Bernoulli numbers.
Math. Comp. 48 (177), pp. 341–350.
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Asymptotic estimates of Stirling numbers.
Stud. Appl. Math. 89 (3), pp. 233–243.
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Explicit formulas for the Bernoulli and Euler polynomials and numbers.
Abh. Math. Sem. Univ. Hamburg 61, pp. 175–180.
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On the theory of the Bernoulli polynomials and numbers.
J. Math. Anal. Appl. 104 (2), pp. 309–350.
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Numerical evaluation of exponential integral: Theis well function approximation.
Journal of Hydrology 205 (1-2), pp. 38–51.
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28: 24.21 Software
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§24.21(ii) , , , and
…29: 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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