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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. …$c\left(n\right)$ 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

… ►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

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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).

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4: 27.12 Asymptotic Formulas: Primes

… ►A pseudoprime test is a test that correctly identifies most composite numbers. For example, if $2^n\cancel{\equiv }2(modn)$, then $n$ is composite. … ►A Carmichael number is a composite number $n$ for which $b^n\equiv b(modn)$ for all $b\in \mathbb{N}$. …

5: 26.2 Basic Definitions

… ►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

… ►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 $\lambda =10$.

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$h$		$\mathrm{erfc}\lambda$		$n$
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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

… ►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: 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 $\xi (𝚪)$ 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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External Links:

MathReview (E. A. Boyd), ZentralBlatt

Referenced by:

(9.13.25), (9.13.26), (9.13.28), (9.13.29), (9.13.32), (9.13.33), (9.13.34), §9.13(ii), §9.13(ii).

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