# compositions

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## 1—10 of 11 matching pages

##### 1: 26.11 Integer Partitions: Compositions

###### §26.11 Integer Partitions: Compositions

►A*composition*is an integer partition in which order is taken into account. …$c\left(n\right)$ denotes the number of compositions of $n$, and ${c}_{m}\left(n\right)$ is the number of compositions into

*exactly*$m$ parts. $c(\in T,n)$ is the number of compositions of $n$ with no 1’s, where again $T=\{2,3,4,\mathrm{\dots}\}$. The integer 0 is considered to have one composition consisting of no parts: …

##### 2: 27.19 Methods of Computation: Factorization

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►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.
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##### 3: 27.22 Software

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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

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►A

*pseudoprime test*is a test that correctly identifies most composite numbers. For example, if ${2}^{n}\overline{)\equiv}2\phantom{\rule{0.949em}{0ex}}(modn)$, then $n$ is composite. … ►A*Carmichael number*is a composite number $n$ for which ${b}^{n}\equiv b\phantom{\rule{0.949em}{0ex}}(modn)$ for all $b\in \mathbb{N}$. …##### 5: 26.2 Basic Definitions

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►Given a finite set $S$ with permutation $\sigma $, a

*cycle*is an ordered equivalence class of elements of $S$ where $j$ is equivalent to $k$ if there exists an $\mathrm{\ell}=\mathrm{\ell}(j,k)$ such that $j={\sigma}^{\mathrm{\ell}}(k)$, where ${\sigma}^{1}=\sigma $ and ${\sigma}^{\mathrm{\ell}}$ is the composition of $\sigma $ with ${\sigma}^{\mathrm{\ell}-1}$. …##### 6: 3.5 Quadrature

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*composite trapezoidal rule*is … ►Then the*elementary Simpson’s rule*is … ►Then the*composite Simpson’s rule*is … ► … ► …##### 7: 31.2 Differential Equations

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###### Composite Transformations

►There are $8\cdot 24=192$ automorphisms of equation (31.2.1) by compositions of $F$-homotopic and homographic transformations. …##### 8: 26.13 Permutations: Cycle Notation

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►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):
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##### 9: 21.5 Modular Transformations

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►The modular transformations form a group under the composition of such transformations, the

*modular group*, which is generated by simpler transformations, for which $\xi (\mathbf{\Gamma})$ is determinate: …##### 10: Bibliography R

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Composite approximations to the solutions of the Orr-Sommerfeld equation.
Studies in Appl. Math. 51, pp. 341–368.
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