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1: 26.11 Integer Partitions: Compositions
§26.11 Integer Partitions: Compositions
A composition is an integer partition in which order is taken into account. … c ( n ) denotes the number of compositions of n , and c m ( n ) is the number of compositions into exactly m parts. c ( T , n ) is the number of compositions of n with no 1’s, where again T = { 2 , 3 , 4 , } . The integer 0 is considered to have one composition consisting of no parts: …
2: 27.19 Methods of Computation: Factorization
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. …
3: 27.22 Software
  • Mathematica. PrimeQ combines strong pseudoprime tests for the bases 2 and 3 and a Lucas pseudoprime test. No known composite numbers pass these three tests, and Bleichenbacher (1996) has shown that this combination of tests proves primality for integers below 10 16 . Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard p - 1 , and then cfrac after trial division. See §27.19. ecm is available also, and the Multiple Polynomial Quadratic sieve is expected in a future release.

    For additional Mathematica routines for factorization and primality testing, including several different pseudoprime tests, see Bressoud and Wagon (2000).

  • 4: 27.12 Asymptotic Formulas: Primes
    A pseudoprime test is a test that correctly identifies most composite numbers. For example, if 2 n 2 ( mod n ) , then n is composite. … A Carmichael number is a composite number n for which b n b ( mod n ) for all b . …
    5: 3.5 Quadrature
    The composite trapezoidal rule is … Then the elementary Simpson’s rule is … Then the composite Simpson’s rule is …
    Table 3.5.20: Composite trapezoidal rule for the integral (3.5.45) with λ = 10 .
    h erfc λ n
    6: 26.2 Basic Definitions
    Given a finite set S with permutation σ , a cycle is an ordered equivalence class of elements of S where j is equivalent to k if there exists an = ( j , k ) such that j = σ ( k ) , where σ 1 = σ and σ is the composition of σ with σ - 1 . …
    7: 31.2 Differential Equations
    Composite Transformations
    There are 8 24 = 192 automorphisms of equation (31.2.1) by compositions of F -homotopic and homographic transformations. …
    8: 26.13 Permutations: Cycle Notation
    A permutation that consists of a single cycle of length k can be written as the composition of k - 1 two-cycles (read from right to left): …
    9: Bibliography R
  • W. H. Reid (1972) Composite approximations to the solutions of the Orr-Sommerfeld equation. Studies in Appl. Math. 51, pp. 341–368.
  • 10: 21.5 Modular Transformations
    The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which ξ ( Γ ) is determinate: …