# pseudorandom numbers

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##### 1: 24.1 Special Notation

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###### 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

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►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: 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*, ${2}^{{2}^{n}}+1$. …

##### 5: 26.11 Integer Partitions: Compositions

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$c\left(n\right)$ 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 $T=\{2,3,4,\mathrm{\dots}\}$. … ►
26.11.1
$$c\left(0\right)=c(\in T,0)=1.$$

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►The *Fibonacci numbers*are determined recursively by … ►Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).##### 6: 26.6 Other Lattice Path Numbers

###### §26.6 Other Lattice Path Numbers

… ►###### Delannoy Number $D(m,n)$

… ►###### Motzkin Number $M(n)$

… ►###### Narayana Number $N(n,k)$

… ►###### §26.6(iv) Identities

…##### 7: 24.15 Related Sequences of Numbers

###### §24.15 Related Sequences of Numbers

►###### §24.15(i) Genocchi Numbers

… ►###### §24.15(ii) Tangent Numbers

… ►###### §24.15(iii) Stirling Numbers

… ►###### §24.15(iv) Fibonacci and Lucas Numbers

…##### 8: 26.5 Lattice Paths: Catalan Numbers

###### §26.5 Lattice Paths: Catalan Numbers

►###### §26.5(i) Definitions

► $C\left(n\right)$ is the Catalan number. … ►###### §26.5(ii) Generating Function

… ►###### §26.5(iii) Recurrence Relations

…##### 9: 26.14 Permutations: Order Notation

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►As an example, $35247816$ is an element of ${\U0001d516}_{8}.$ The

*inversion number*is the number of pairs of elements for which the larger element precedes the smaller: … ► ►The*Eulerian number*, denoted $$, is the number of permutations in ${\U0001d516}_{n}$ with exactly $k$ descents. …The Eulerian number $$ is equal to the number of permutations in ${\U0001d516}_{n}$ with exactly $k$ excedances. … ►