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31: 3.1 Arithmetics and Error Measures

… ►

§3.1(i) Floating-Point Arithmetic

… ► ►The set of machine numbers ${ℝ}_{\mathrm{fl}}$ is the union of $0$ and the set … ►Let $x$ be any positive number with … ►Negative numbers $x$ are rounded in the same way as $-x$. …

32: 23.20 Mathematical Applications

… ►Given $P$, calculate $2P$, $4P$, $8P$ by doubling as above. …If any of $2P$, $4P$, $8P$ is not an integer, then the point has infinite order. Otherwise observe any equalities between $P$, $2P$, $4P$, $8P$, and their negatives. The order of a point (if finite and not already determined) can have only the values 3, 5, 6, 7, 9, 10, or 12, and so can be found from $2P=-P$, $4P=-P$, $4P=-2P$, $8P=P$, $8P=-P$, $8P=-2P$, or $8P=-4P$. … ►

§23.20(v) Modular Functions and Number Theory

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33: 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, $2^{2^n}+1$. …

34: 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. $c(\in T,n)$ 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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Symbols:

$\in$: element of, $c\left(n\right)$: number of compositions of $n$ , $c(\mathrm{condition},n)$: restricted number of compositions of $n$ and $T$: set

Permalink:

http://dlmf.nist.gov/26.11.E1

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Annotations for §26.11 and Ch.26

… ►The Fibonacci numbers are determined recursively by … ►Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).

35: 13.9 Zeros

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§13.9(i) Zeros of $M(a,b,z)$

… ►When $a,b\in ℝ$ the number of real zeros is finite. Let $p(a,b)$ be the number of positive zeros. …The number of negative real zeros $n(a,b)$ is given by … ►

§13.9(ii) Zeros of $U(a,b,z)$

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36: 26.2 Basic Definitions

… ►The total number of partitions of $n$ is denoted by $p\left(n\right)$. … ►

Table 26.2.1: Partitions $p\left(n\right)$.

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$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
…
8	22	25	1958	42	53174
…

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37: 26.12 Plane Partitions

… ► … ►Then the number of plane partitions in $B(r,s,t)$ is … ►The number of symmetric plane partitions in $B(r,r,t)$ is … ►The number of cyclically symmetric plane partitions in $B(r,r,r)$ is … ►The number of descending plane partitions in $B(r,r,r)$ is …

38: 24.8 Series Expansions

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24.8.6 $$B_{4n+2}=(8n+4)\sum _{k=1}^{\mathrm{\infty }}\frac{k^{4n+1}}{{\mathrm{e}}^{2\pi k}-1},$$ $n=1,2,\mathrm{\dots }$,

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $k$: integer and $n$: integer

Referenced by:

§24.8(ii)

Permalink:

http://dlmf.nist.gov/24.8.E6

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Annotations for §24.8(ii), §24.8 and Ch.24

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24.8.7 $$B_{2n}=\frac{{(-1)}^{n+1}4n}{2^{2n}-1}\sum _{k=1}^{\mathrm{\infty }}\frac{k^{2n-1}}{{\mathrm{e}}^{\pi k}+{(-1)}^{k+n}},$$ $n=2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $k$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/24.8.E7

Encodings:

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See also:

Annotations for §24.8(ii), §24.8 and Ch.24

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24.8.8 $$\frac{B_{2n}}{4n}\left({\alpha }^n-{(-\beta )}^n\right)={\alpha }^n\sum _{k=1}^{\mathrm{\infty }}\frac{k^{2n-1}}{{\mathrm{e}}^{2\alpha k}-1}-{(-\beta )}^n\sum _{k=1}^{\mathrm{\infty }}\frac{k^{2n-1}}{{\mathrm{e}}^{2\beta k}-1},$$ $n=2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\mathrm{e}$: base of natural logarithm, $k$: integer, $n$: integer, $\alpha$: parameter and $\beta$: parameter

Referenced by:

§24.8(ii)

Permalink:

http://dlmf.nist.gov/24.8.E8

Encodings:

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Annotations for §24.8(ii), §24.8 and Ch.24

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24.8.9 $$E_{2n}={(-1)}^n\sum _{k=1}^{\mathrm{\infty }}\frac{k^{2n}}{\cosh \left(\frac{1}{2}\pi k\right)}-4\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{(2k+1)}^{2n}}{{\mathrm{e}}^{2\pi (2k+1)}-1},$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $k$: integer and $n$: integer

Keywords:

Euler polynomials, infinite series expansions, other

Permalink:

http://dlmf.nist.gov/24.8.E9

Encodings:

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Annotations for §24.8(ii), §24.8 and Ch.24

39: DLMF Project News

error generating summary

40: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ► $\left(\genfrac{}{}{0.0pt}{}{n}{n_1,n_2,\mathrm{\dots },n_k}\right)$ is the number of ways of placing $n=n_1+n_2+\mathrm{\cdots }+n_k$ distinct objects into $k$ labeled boxes so that there are $n_j$ objects in the $j$th box. It is also the number of $k$-dimensional lattice paths from $(0,0,\mathrm{\dots },0)$ to $(n_1,n_2,\mathrm{\dots },n_k)$. … ► $M_2$ is the number of permutations of $\{1,2,\mathrm{\dots },n\}$ with $a_1$ cycles of length 1, $a_2$ cycles of length 2, $\mathrm{\dots }$, and $a_n$ cycles of length $n$: ►

26.4.7 $$M_2=\frac{n!}{1^{a_1}(a_1!){\mathrm{\hspace{0.17em}2}}^{a_2}(a_2!)\mathrm{\cdots }{n}^{a_n}(a_n!)}.$$

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Symbols:

$!$: factorial (as in $n!$), $n$: nonnegative integer, $a_n$: nonnegative integers and $M_2$: number of permutations

Referenced by:

§26.13

Permalink:

http://dlmf.nist.gov/26.4.E7

Encodings:

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Annotations for §26.4(i), §26.4 and Ch.26

… ► $M_3$ is the number of set partitions of $\{1,2,\mathrm{\dots },n\}$ with $a_1$ subsets of size 1, $a_2$ subsets of size 2, $\mathrm{\dots }$, and $a_n$ subsets of size $n$: …