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1: 27.19 Methods of Computation: Factorization
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. … Type I probabilistic algorithms include the Brent–Pollard rho algorithm (also called Monte Carlo method), the Pollard p - 1 algorithm, and the Elliptic Curve Method (ecm). … Type II probabilistic algorithms for factoring n rely on finding a pseudo-random pair of integers ( x , y ) that satisfy x 2 y 2 ( mod n ) . …As of January 2009 the snfs holds the record for the largest integer that has been factored by a Type II probabilistic algorithm, a 307-digit composite integer. …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. …
2: 18.36 Miscellaneous Polynomials
§18.36(i) Jacobi-Type Polynomials
3: 10.64 Integral Representations
Schläfli-Type Integrals
4: Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate
Jacobian elliptic functions arise as solutions to certain nonlinear Schrödinger equations, which model many types of wave propagation phenomena. …
5: How to Cite
Item Type Ref. Number Permalink
6: 18.31 Bernstein–Szegő Polynomials
The Bernstein–Szegő polynomials { p n ( x ) } , n = 0 , 1 , , are orthogonal on ( - 1 , 1 ) with respect to three types of weight function: ( 1 - x 2 ) - 1 2 ( ρ ( x ) ) - 1 , ( 1 - x 2 ) 1 2 ( ρ ( x ) ) - 1 , ( 1 - x ) 1 2 ( 1 + x ) - 1 2 ( ρ ( x ) ) - 1 . …
7: 19.38 Approximations
Approximations of the same type for K ( k ) and E ( k ) for 0 < k 1 are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. …
8: 20.8 Watson’s Expansions
This reference and Bellman (1961, pp. 46–47) include other expansions of this type.
9: 16.6 Transformations of Variable
16.6.2 F 2 3 ( a , 2 b - a - 1 , 2 - 2 b + a b , a - b + 3 2 ; z 4 ) = ( 1 - z ) - a F 2 3 ( 1 3 a , 1 3 a + 1 3 , 1 3 a + 2 3 b , a - b + 3 2 ; - 27 z 4 ( 1 - z ) 3 ) .
For Kummer-type transformations of F 2 2 functions see Miller (2003) and Paris (2005a), and for further transformations see Erdélyi et al. (1953a, §4.5), Miller and Paris (2011), Choi and Rathie (2013) and Wang and Rathie (2013).
10: 12.12 Integrals
Nicholson-type Integral