# pseudorandom numbers

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##### 1: 24.1 Special Notation
###### Bernoulli Numbers and Polynomials
The origin of the notation $B_{n}$, $B_{n}\left(x\right)$, is not clear. …
###### Euler Numbers and Polynomials
Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations $E_{n}$, $E_{n}\left(x\right)$, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
##### 2: 27.17 Other Applications
###### §27.17 Other Applications
Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. Congruences are used in constructing perpetual calendars, splicing telephone cables, scheduling round-robin tournaments, devising systematic methods for storing computer files, and generating pseudorandom numbers. … There are also applications of number theory in many diverse areas, including physics, biology, chemistry, communications, and art. …
##### 3: 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. … As of January 2009 the largest prime factors found by these methods are a 19-digit prime for Brent–Pollard rho, a 58-digit prime for Pollard $p-1$, and a 67-digit prime for ecm. … These algorithms include the Continued Fraction Algorithm (cfrac), the Multiple Polynomial Quadratic Sieve (mpqs), the General Number Field Sieve (gnfs), and the Special Number Field Sieve (snfs). …The snfs can be applied only to numbers that are very close to a power of a very small base. 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. …
##### 4: 26.11 Integer Partitions: Compositions
$c\left(n\right)$ 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 $T=\{2,3,4,\ldots\}$. …
26.11.1 $c\left(0\right)=c\left(\in\!T,0\right)=1.$
The Fibonacci numbers are determined recursively by … Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).
##### 5: 27.18 Methods of Computation: Primes
###### §27.18 Methods of Computation: Primes
An overview of methods for precise counting of the number of primes not exceeding an arbitrary integer $x$ is given in Crandall and Pomerance (2005, §3.7). …An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). … These algorithms are used for testing primality of Mersenne numbers, $2^{n}-1$, and Fermat numbers, $2^{2^{n}}+1$. …
##### 8: 26.5 Lattice Paths: Catalan Numbers
###### §26.5(i) Definitions
$C\left(n\right)$ is the Catalan number. …
##### 9: 26.14 Permutations: Order Notation
As an example, $35247816$ is an element of $\mathfrak{S}_{8}.$ The inversion number is the number of pairs of elements for which the larger element precedes the smaller: … The Eulerian number, denoted $\genfrac{<}{>}{0.0pt}{}{n}{k}$, is the number of permutations in $\mathfrak{S}_{n}$ with exactly $k$ descents. …The Eulerian number $\genfrac{<}{>}{0.0pt}{}{n}{k}$ is equal to the number of permutations in $\mathfrak{S}_{n}$ with exactly $k$ excedances. …