# prime numbers

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##### 1: 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 …
27.3.5 $d\left(n\right)=\prod_{r=1}^{\nu\left(n\right)}(1+a_{r}),$
##### 2: 27.12 Asymptotic Formulas: Primes
###### §27.12 Asymptotic Formulas: Primes
$\pi\left(x\right)$ is the number of primes less than or equal to $x$. …
$\pi\left(x\right)-\operatorname{li}\left(x\right)$ changes sign infinitely often as $x\to\infty$; see Littlewood (1914), Bays and Hudson (2000). … The largest known prime (2018) is the Mersenne prime $2^{82,589,933}-1$. …
##### 3: 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.) …There is great interest in the function $\pi\left(x\right)$ that counts the number of primes not exceeding $x$. … (See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) …
##### 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). … Oliveira e Silva has calculated $\pi\left(x\right)$ for $x=10^{23}$, using the combinatorial methods of Lagarias et al. (1985) and Deléglise and Rivat (1996); see Oliveira e Silva (2006). An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). … For small values of $n$, primality is proven by showing that $n$ is not divisible by any prime not exceeding $\sqrt{n}$. …
27.9.2 $(2|p)=(-1)^{(p^{2}-1)/8}.$
If $p,q$ are distinct odd primes, then the quadratic reciprocity law states that
27.9.3 $(p|q)(q|p)=(-1)^{(p-1)(q-1)/4}.$
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 $(n|1)=1$. …
##### 6: 27.21 Tables
###### §27.21 Tables
Bressoud and Wagon (2000, pp. 103–104) supplies tables and graphs that compare $\pi\left(x\right),\ifrac{x}{\ln x}$, and $\operatorname{li}\left(x\right)$. …
##### 7: 27.6 Divisor Sums
27.6.2 $\sum_{d\mathbin{|}n}\mu\left(d\right)f(d)=\prod_{p\mathbin{|}n}(1-f(p)),$ $n>1$.
##### 8: 27.1 Special Notation
 $d,k,m,n$ positive integers (unless otherwise indicated). … prime numbers (or primes): integers ($>1$) with only two positive integer divisors, $1$ and the number itself. …
##### 9: 27.4 Euler Products and Dirichlet Series
27.4.1 $\sum_{n=1}^{\infty}f(n)=\prod_{p}\left(1+\sum_{r=1}^{\infty}f(p^{r})\right),$
27.4.2 $\sum_{n=1}^{\infty}f(n)=\prod_{p}(1-f(p))^{-1}.$
27.4.3 $\zeta\left(s\right)=\sum_{n=1}^{\infty}n^{-s}=\prod_{p}(1-p^{-s})^{-1},$ $\Re s>1$.
27.4.9 $\sum_{n=1}^{\infty}2^{\nu\left(n\right)}n^{-s}=\frac{(\zeta\left(s\right))^{2}% }{\zeta\left(2s\right)},$ $\Re s>1$,
##### 10: 27.16 Cryptography
###### §27.16 Cryptography
The primes are kept secret but their product $n=pq$, an 800-digit number, is made public. …