# composite

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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. $c\left(\in\!T,n\right)$ is the number of compositions of $n$ with no 1’s, where again $T=\{2,3,4,\ldots\}$. 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 $2^{n}\not\equiv 2\pmod{n}$, then $n$ is composite. … A Carmichael number is a composite number $n$ for which $b^{n}\equiv b\pmod{n}$ 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 $\ell=\ell(j,k)$ such that $j=\sigma^{\ell}(k)$, where $\sigma^{1}=\sigma$ and $\sigma^{\ell}$ is the composition of $\sigma$ with $\sigma^{\ell-1}$. …
There are $8\cdot 24=192$ automorphisms of equation (31.2.1) by compositions of $F$-homotopic and homographic transformations. …
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): …
The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which $\xi(\boldsymbol{{\Gamma}})$ is determinate: …