Prime Numbers
Generate a list of prime numbers less or equal n, resp. between
n1 and n2.
primes(n)
n |
nonnegative integer greater than 1. |
The list of prime numbers up to n is generated using the "sieve of
Erasthostenes". This approach is reasonably fast, but may require a lot of
main memory when n is large.
In double precision arithmetic integers are represented exactly only up to 2^53 - 1, therefore this is the maximal allowed value.
vector of integers representing prime numbers
primes(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
# --------------------------------------
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