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1: 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). … ►The Sieve of Eratosthenes (Crandall and Pomerance (2005, §3.2)) generates a list of all primes below a given bound. … ►For small values of $n$, primality is proven by showing that $n$ is not divisible by any prime not exceeding $\sqrt{n}$. … ►The ECPP (Elliptic Curve Primality Proving) algorithm handles primes with over 20,000 digits. …

2: 27.12 Asymptotic Formulas: Primes

§27.12 Asymptotic Formulas: Primes

… ►

Prime Number Theorem

… ►The number of such primes not exceeding $x$ is … ►A Mersenne prime is a prime of the form $2^p-1$. The largest known prime (2018) is the Mersenne prime $2^{82,589,933}-1$. …

3: 27.16 Cryptography

§27.16 Cryptography

… ►For example, a code maker chooses two large primes $p$ and $q$ of about 400 decimal digits each. Procedures for finding such primes require very little computer time. … ►Choose a prime $r$ that does not divide either $p-1$ or $q-1$. Like $n$, the prime $r$ is made public. …

4: 29.10 Lamé Functions with Imaginary Periods

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29.10.2 $$z^{\prime }=\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),$$

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Symbols:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.10 and Ch.29

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29.10.3 $$\frac{d^{2}w}{{d{z}^{\prime }}^2}+(h^{\prime }-\nu (\nu +1)k_{}^{\prime }{}_{}{}^2{\mathrm{sn}}^2(z^{\prime },k^{\prime }))w=0.$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $h$: real parameter, $k$: real parameter, $\nu$: real parameter and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/29.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §29.10 and Ch.29

… ►

${\mathrm{𝐸𝑐}}_{\nu }^{2m}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k_{}^{\prime }{}_{}{}^2),$

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${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k_{}^{\prime }{}_{}{}^2),$

… ►The first and the fourth functions have period $2\mathrm{i}{K}^{\prime }$; the second and the third have period $4\mathrm{i}{K}^{\prime }$. …

5: 27.9 Quadratic Characters

§27.9 Quadratic Characters

►For an odd prime $p$, the Legendre symbol $(n|p)$ is defined as follows. … ►If $p,q$ are distinct odd primes, then the quadratic reciprocity law states that … ►If an odd integer $P$ has prime factorization $P={\prod }_{r=1}^{\nu \left(n\right)}{p}_r^{a_r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)={\prod }_{r=1}^{\nu \left(n\right)}{(n|p_r)}^{a_r}$, with $(n|1)=1$. …Both (27.9.1) and (27.9.2) are valid with $p$ replaced by $P$; the reciprocity law (27.9.3) holds if $p,q$ are replaced by any two relatively prime odd integers $P,Q$.

6: 27.2 Functions

… ►where $p_1,p_2,\mathrm{\dots },p_{\nu \left(n\right)}$ are the distinct prime factors of $n$, each exponent $a_r$ is positive, and $\nu \left(n\right)$ is the number of distinct primes dividing $n$. ($\nu \left(1\right)$ is defined to be 0.) …Tables of primes (§27.21) reveal great irregularity in their distribution. … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ►

§27.2(ii) Tables

…

7: 22.7 Landen Transformations

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22.7.1 $$k_1=\frac{1-k^{\prime }}{1+k^{\prime }},$$

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Defines:

$k_1$: change of variable (locally)

Symbols:

$k^{\prime }$: complementary modulus

A&S Ref:

16.12.1

Referenced by:

§22.20(iii)

Permalink:

http://dlmf.nist.gov/22.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.7(i), §22.7 and Ch.22

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$k_2^{\prime }={\displaystyle \frac{1-k}{1+k}},$

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22.7.6 $$\mathrm{sn}(z,k)=\frac{(1+k_2^{\prime })\mathrm{sn}(z/(1+k_2^{\prime }),k_2)\mathrm{cn}(z/(1+k_2^{\prime }),k_2)}{\mathrm{dn}(z/(1+k_2^{\prime }),k_2)},$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus, $k_2$: change of variable and $k_2^{\prime }$: change of variable

A&S Ref:

16.14.2

Permalink:

http://dlmf.nist.gov/22.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.7(ii), §22.7 and Ch.22

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22.7.7 $$\mathrm{cn}(z,k)=\frac{(1+k_2^{\prime })({\mathrm{dn}}^2(z/(1+k_2^{\prime }),k_2)-k_2^{\prime })}{k_2^2\mathrm{dn}(z/(1+k_2^{\prime }),k_2)},$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus, $k_2$: change of variable and $k_2^{\prime }$: change of variable

A&S Ref:

16.14.3

Permalink:

http://dlmf.nist.gov/22.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.7(ii), §22.7 and Ch.22

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22.7.8 $$\mathrm{dn}(z,k)=\frac{(1-k_2^{\prime })({\mathrm{dn}}^2(z/(1+k_2^{\prime }),k_2)+k_2^{\prime })}{k_2^2\mathrm{dn}(z/(1+k_2^{\prime }),k_2)}.$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus, $k_2$: change of variable and $k_2^{\prime }$: change of variable

A&S Ref:

16.14.4

Permalink:

http://dlmf.nist.gov/22.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.7(ii), §22.7 and Ch.22

…

8: 27.1 Special Notation

… ► ►►►►►►

$d,k,m,n$	positive integers (unless otherwise indicated).
…
$\left(m,n\right)$	greatest common divisor of $m,n$. If $\left(m,n\right)=1$, $m$ and $n$ are called relatively prime, or coprime.
…
${\sum }_{\left(m,n\right)=1}$	sum taken over $m$, $1\le m\le n$ and $m$ relatively prime to $n$.
$p,p_1,p_2,\mathrm{\dots }$	prime numbers (or primes): integers ($>1$) with only two positive integer divisors, $1$ and the number itself.
${\sum }_p$, ${\prod }_p$	sum, product extended over all primes.
…

9: 27.3 Multiplicative Properties

… ►Except for $\nu \left(n\right)$, $\mathrm{\Lambda }\left(n\right)$, $p_n$, and $\pi \left(x\right)$, the functions in §27.2 are multiplicative, which means $f(1)=1$ and … ►If $f$ is multiplicative, then the values $f(n)$ for $n>1$ are determined by the values at the prime powers. … ►

27.3.2 $$f(n)=\prod _{r=1}^{\nu \left(n\right)}f(p_r^{a_r}).$$

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Symbols:

$\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers, $a_r$: positive exponent and $f(n)$: multiplicative function

Referenced by:

§27.20, §27.20, §27.3

Permalink:

http://dlmf.nist.gov/27.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

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27.3.5 $$d\left(n\right)=\prod _{r=1}^{\nu \left(n\right)}(1+a_r),$$

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Symbols:

$d_k\left(n\right)$: divisor function, $\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer and $a_r$: positive exponent

Permalink:

http://dlmf.nist.gov/27.3.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

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27.3.10 $$f(n)=\prod _{r=1}^{\nu \left(n\right)}{\left(f(p_r)\right)}^{a_r}.$$

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Symbols:

$\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers, $a_r$: positive exponent and $f(n)$: multiplicative function

Referenced by:

§27.20, §27.20

Permalink:

http://dlmf.nist.gov/27.3.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

10: 10.58 Zeros

… ►For $n\ge 0$ the $m$th positive zeros of ${𝗃}_n\left(x\right)$, ${𝗃}_n^{\prime }\left(x\right)$, ${𝗒}_n\left(x\right)$, and ${𝗒}_n^{\prime }\left(x\right)$ are denoted by $a_{n,m}$, $a_{n,m}^{\prime }$, $b_{n,m}$, and $b_{n,m}^{\prime }$, respectively, except that for $n=0$ we count $x=0$ as the first zero of ${𝗃}_0^{\prime }\left(x\right)$. … ►

${𝗃}_n^{\prime }\left(a_{n,m}\right)=\sqrt{{\displaystyle \frac{\pi }{2j_{n+\frac{1}{2},m}}}}{J}_{n+\frac{1}{2}}^{\prime }\left(j_{n+\frac{1}{2},m}\right),$

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${𝗒}_n^{\prime }\left(b_{n,m}\right)=\sqrt{{\displaystyle \frac{\pi }{2y_{n+\frac{1}{2},m}}}}{Y}_{n+\frac{1}{2}}^{\prime }\left(y_{n+\frac{1}{2},m}\right).$

… ►For some properties of $a_{n,m}^{\prime }$ and $b_{n,m}^{\prime }$, including asymptotic expansions, see Olver (1960, pp. xix–xxi). …