# primes

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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
###### 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: 29.10 Lamé Functions with Imaginary Periods
29.10.2 $z^{\prime}=\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),$
29.10.3 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z^{\prime}}^{2}}+(h^{\prime}-\nu(\nu+1){k^% {\prime}}^{2}{\operatorname{sn}}^{2}\left(z^{\prime},k^{\prime}\right))w=0.$
$\mathit{Ec}^{2m}_{\nu}\left(\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),{k^{\prime}% }^{2}\right),$
$\mathit{Ec}^{2m+1}_{\nu}\left(\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),{k^{% \prime}}^{2}\right),$
The first and the fourth functions have period $2\mathrm{i}{K^{\prime}}$; the second and the third have period $4\mathrm{i}{K^{\prime}}$. …
##### 4: 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 $P=\prod_{r=1}^{\nu\left(n\right)}p^{a_{r}}_{r}$, 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 $P,Q$.
##### 5: 27.2 Functions
where $p_{1},p_{2},\dots,p_{\nu\left(n\right)}$ 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 primes27.21) reveal great irregularity in their distribution. … (See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) …
##### 6: 22.7 Landen Transformations
22.7.1 $k_{1}=\frac{1-k^{\prime}}{1+k^{\prime}},$
$k_{2}^{\prime}=\frac{1-k}{1+k},$
22.7.6 $\operatorname{sn}\left(z,k\right)=\frac{(1+k_{2}^{\prime})\operatorname{sn}% \left(z/(1+k_{2}^{\prime}),k_{2}\right)\operatorname{cn}\left(z/(1+k_{2}^{% \prime}),k_{2}\right)}{\operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right% )},$
22.7.7 $\operatorname{cn}\left(z,k\right)=\frac{(1+k_{2}^{\prime})({\operatorname{dn}}% ^{2}\left(z/(1+k_{2}^{\prime}),k_{2}\right)-k_{2}^{\prime})}{k_{2}^{2}% \operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right)},$
22.7.8 $\operatorname{dn}\left(z,k\right)=\frac{(1-k_{2}^{\prime})({\operatorname{dn}}% ^{2}\left(z/(1+k_{2}^{\prime}),k_{2}\right)+k_{2}^{\prime})}{k_{2}^{2}% \operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right)}.$
##### 7: 27.1 Special Notation
 $d,k,m,n$ positive integers (unless otherwise indicated). … greatest common divisor of $m,n$. If $\left(m,n\right)=1$, $m$ and $n$ are called relatively prime, or coprime. … sum taken over $m$, $1\leq m\leq n$ and $m$ relatively prime to $n$. prime numbers (or primes): integers ($>1$) with only two positive integer divisors, $1$ and the number itself. sum, product extended over all primes. …
##### 8: 27.3 Multiplicative Properties
Except for $\nu\left(n\right)$, $\Lambda\left(n\right)$, $p_{n}$, 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. …
27.3.5 $d\left(n\right)=\prod_{r=1}^{\nu\left(n\right)}(1+a_{r}),$
##### 9: 10.58 Zeros
For $n\geq 0$ the $m$th positive zeros of $\mathsf{j}_{n}\left(x\right)$, $\mathsf{j}_{n}'\left(x\right)$, $\mathsf{y}_{n}\left(x\right)$, and $\mathsf{y}_{n}'\left(x\right)$ are denoted by $a_{n,m}$, $a^{\prime}_{n,m}$, $b_{n,m}$, and $b^{\prime}_{n,m}$, respectively, except that for $n=0$ we count $x=0$ as the first zero of $\mathsf{j}_{0}'\left(x\right)$. …
$\mathsf{j}_{n}'\left(a_{n,m}\right)=\sqrt{\frac{\pi}{2j_{n+\frac{1}{2},m}}}J_{% n+\frac{1}{2}}'\left(j_{n+\frac{1}{2},m}\right),$
$\mathsf{y}_{n}'\left(b_{n,m}\right)=\sqrt{\frac{\pi}{2y_{n+\frac{1}{2},m}}}Y_{% n+\frac{1}{2}}'\left(y_{n+\frac{1}{2},m}\right).$
For some properties of $a^{\prime}_{n,m}$ and $b^{\prime}_{n,m}$, including asymptotic expansions, see Olver (1960, pp. xix–xxi). …
##### 10: 16.15 Integral Representations and Integrals
16.15.1 ${F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=\frac{\Gamma\left(% \gamma\right)}{\Gamma\left(\alpha\right)\Gamma\left(\gamma-\alpha\right)}\int_% {0}^{1}\frac{u^{\alpha-1}(1-u)^{\gamma-\alpha-1}}{(1-ux)^{\beta}(1-uy)^{\beta^% {\prime}}}\mathrm{d}u,$ $\Re\alpha>0$, $\Re\left(\gamma-\alpha\right)>0$,
16.15.2 ${F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y\right)=% \frac{\Gamma\left(\gamma\right)\Gamma\left(\gamma^{\prime}\right)}{\Gamma\left% (\beta\right)\Gamma\left(\beta^{\prime}\right)\Gamma\left(\gamma-\beta\right)% \Gamma\left(\gamma^{\prime}-\beta^{\prime}\right)}\int_{0}^{1}\!\!\!\int_{0}^{% 1}\frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u)^{\gamma-\beta-1}(1-v)^{\gamma^{% \prime}-\beta^{\prime}-1}}{(1-ux-vy)^{\alpha}}\mathrm{d}u\mathrm{d}v,$ $\Re\gamma>\Re\beta>0$, $\Re\gamma^{\prime}>\Re\beta^{\prime}>0$,
16.15.3 ${F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime};\gamma;x,y\right)=% \frac{\Gamma\left(\gamma\right)}{\Gamma\left(\beta\right)\Gamma\left(\beta^{% \prime}\right)\Gamma\left(\gamma-\beta-\beta^{\prime}\right)}\iint_{\Delta}% \frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u-v)^{\gamma-\beta-\beta^{\prime}-1}}{% (1-ux)^{\alpha}(1-vy)^{\alpha^{\prime}}}\mathrm{d}u\mathrm{d}v,$ $\Re\left(\gamma-\beta-\beta^{\prime}\right)>0$, $\Re\beta>0$, $\Re\beta^{\prime}>0$,
16.15.4 ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x(1-y),y(1-x)\right)=\frac{% \Gamma\left(\gamma\right)\Gamma\left(\gamma^{\prime}\right)}{\Gamma\left(% \alpha\right)\Gamma\left(\beta\right)\Gamma\left(\gamma-\alpha\right)\Gamma% \left(\gamma^{\prime}-\beta\right)}\int_{0}^{1}\!\!\!\int_{0}^{1}\frac{u^{% \alpha-1}v^{\beta-1}(1-u)^{\gamma-\alpha-1}(1-v)^{\gamma^{\prime}-\beta-1}}{(1% -ux)^{\gamma+\gamma^{\prime}-\alpha-1}(1-vy)^{\gamma+\gamma^{\prime}-\beta-1}(% 1-ux-vy)^{\alpha+\beta-\gamma-\gamma^{\prime}+1}}\mathrm{d}u\mathrm{d}v,$ $\Re\gamma>\Re\alpha>0$, $\Re\gamma^{\prime}>\Re\beta>0$.