Primes {numbers}

Prime Numbers

Description

Eratosthenes resp. Atkin sieve methods to generate a list of prime numbers less or equal n, resp. between n1 and n2.

Usage

  Primes(n1, n2 = NULL)

  atkin_sieve(n)

Arguments

n, n1, n2

natural numbers with n1 <= n2.

Details

The list of prime numbers up to n is generated using the "sieve of Eratosthenes". This approach is reasonably fast, but may require a lot of main memory when n is large.

Primes computes first all primes up to sqrt(n2) and then applies a refined sieve on the numbers from n1 to n2, thereby drastically reducing the need for storing long arrays of numbers.

The sieve of Atkins is a modified version of the ancient prime number sieve of Eratosthenes. It applies a modulo-sixty arithmetic and requires less memory, but in R is not faster because of a double for-loop.

In double precision arithmetic integers are represented exactly only up to 2^53 - 1, therefore this is the maximal allowed value.

Value

vector of integers representing prime numbers

References

A. Atkin and D. Bernstein (2004), Prime sieves using quadratic forms. Mathematics of Computation, Vol. 73, pp. 1023-1030.

See Also

isPrime, gmp::factorize, pracma::expint1

Examples

Primes(1000)
Primes(1949, 2019)

atkin_sieve(1000)

## Not run: 
##  Appendix:  Logarithmic Integrals and Prime Numbers (C.F.Gauss, 1846)

library('gsl')
# 'European' form of the logarithmic integral
Li <- function(x) expint_Ei(log(x)) - expint_Ei(log(2))

# No. of primes and logarithmic integral for 10^i, i=1..12
i <- 1:12;  N <- 10^i
# piN <- numeric(12)
# for (i in 1:12) piN[i] <- length(primes(10^i))
piN <- c(4, 25, 168, 1229, 9592, 78498, 664579,
         5761455, 50847534, 455052511, 4118054813, 37607912018)
cbind(i, piN, round(Li(N)), round((Li(N)-piN)/piN, 6))

#  i     pi(10^i)      Li(10^i)  rel.err  
# --------------------------------------      
#  1            4            5  0.280109
#  2           25           29  0.163239
#  3          168          177  0.050979
#  4         1229         1245  0.013094
#  5         9592         9629  0.003833
#  6        78498        78627  0.001637
#  7       664579       664917  0.000509
#  8      5761455      5762208  0.000131
#  9     50847534     50849234  0.000033
# 10    455052511    455055614  0.000007
# 11   4118054813   4118066400  0.000003
# 12  37607912018  37607950280  0.000001
# --------------------------------------
## End(Not run)