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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 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: 29.10 Lamé Functions with Imaginary Periods
29.10.3 d 2 w d z 2 + ( h - ν ( ν + 1 ) k 2 sn 2 ( z , k ) ) w = 0 .
Ec ν 2 m ( i ( z - K - i K ) , k 2 ) ,
Ec ν 2 m + 1 ( i ( z - K - i K ) , k 2 ) ,
The first and the fourth functions have period 2 i K ; the second and the third have period 4 i K . …
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 = r = 1 ν ( n ) p r a r , then the Jacobi symbol ( n | P ) is defined by ( n | P ) = r = 1 ν ( n ) ( 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 , , p ν ( n ) are the distinct prime factors of n , each exponent a r is positive, and ν ( n ) is the number of distinct primes dividing n . ( ν ( 1 ) 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).) …
§27.2(ii) Tables
6: 22.7 Landen Transformations
22.7.1 k 1 = 1 - k 1 + k ,
k 2 = 1 - k 1 + k ,
22.7.6 sn ( z , k ) = ( 1 + k 2 ) sn ( z / ( 1 + k 2 ) , k 2 ) cn ( z / ( 1 + k 2 ) , k 2 ) dn ( z / ( 1 + k 2 ) , k 2 ) ,
22.7.7 cn ( z , k ) = ( 1 + k 2 ) ( dn 2 ( z / ( 1 + k 2 ) , k 2 ) - k 2 ) k 2 2 dn ( z / ( 1 + k 2 ) , k 2 ) ,
22.7.8 dn ( z , k ) = ( 1 - k 2 ) ( dn 2 ( z / ( 1 + k 2 ) , k 2 ) + k 2 ) k 2 2 dn ( z / ( 1 + k 2 ) , k 2 ) .
7: 27.1 Special Notation
d , k , m , n positive integers (unless otherwise indicated).
( m , n ) greatest common divisor of m , n . If ( m , n ) = 1 , m and n are called relatively prime, or coprime.
( m , n ) = 1 sum taken over m , 1 m n and m relatively prime to n .
p , p 1 , p 2 , prime numbers (or primes): integers ( > 1 ) with only two positive integer divisors, 1 and the number itself.
p , p sum, product extended over all primes.
8: 27.3 Multiplicative Properties
Except for ν ( n ) , Λ ( n ) , p n , and π ( x ) , 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 ) = r = 1 ν ( n ) f ( p r a r ) .
27.3.5 d ( n ) = r = 1 ν ( n ) ( 1 + a r ) ,
27.3.10 f ( n ) = r = 1 ν ( n ) ( f ( p r ) ) a r .
9: 10.58 Zeros
For n 0 the m th positive zeros of j n ( x ) , j n ( x ) , y n ( x ) , and y n ( x ) are denoted by a n , m , a n , m , b n , m , and b n , m , respectively, except that for n = 0 we count x = 0 as the first zero of j 0 ( x ) . …
j n ( a n , m ) = π 2 j n + 1 2 , m J n + 1 2 ( j n + 1 2 , m ) ,
y n ( b n , m ) = π 2 y n + 1 2 , m Y n + 1 2 ( y n + 1 2 , m ) .
For some properties of a n , m and b n , m , including asymptotic expansions, see Olver (1960, pp. xix–xxi). …
10: 16.13 Appell Functions
16.13.1 F 1 ( α ; β , β ; γ ; x , y ) = m , n = 0 ( α ) m + n ( β ) m ( β ) n ( γ ) m + n m ! n ! x m y n , max ( | x | , | y | ) < 1 ,
16.13.2 F 2 ( α ; β , β ; γ , γ ; x , y ) = m , n = 0 ( α ) m + n ( β ) m ( β ) n ( γ ) m ( γ ) n m ! n ! x m y n , | x | + | y | < 1 ,
16.13.3 F 3 ( α , α ; β , β ; γ ; x , y ) = m , n = 0 ( α ) m ( α ) n ( β ) m ( β ) n ( γ ) m + n m ! n ! x m y n , max ( | x | , | y | ) < 1 ,
16.13.4 F 4 ( α , β ; γ , γ ; x , y ) = m , n = 0 ( α ) m + n ( β ) m + n ( γ ) m ( γ ) n m ! n ! x m y n , | x | + | y | < 1 .
Here and elsewhere it is assumed that neither of the bottom parameters γ and γ is a nonpositive integer. …