VeryLargeIntegers (version 0.2.1)

21. Pi function: Pi Function Approximation for vli Objects

Description

Pi function approximation for vli (Very Large Integers) objects. It is also called "Prime-counting function". Given a positive integer x, the Pi function returns the number of primes up to x.

Usage

Pi(x)
# S3 method for default
Pi(x)
# S3 method for numeric
Pi(x)
# S3 method for vli
Pi(x)

Value

number of primes up to x; object of class vli

Arguments

x

positive integer; vli class object or 32 bits integer

Author

Javier Leiva Cuadrado

Details

The implemented algorithm is based in the fact that x/log(x) is asymptotically equal to Pi(x), also known as "Prime Number Theorem".

Closer approximations could be implemented by using the Logarithmic Integral Function. The function countprimes of the present package is another way to get a better approximation (in return for a less efficient computation) of Pi(x). Alhought the algorithm is not deterministic, it is based in the Miller-Rabin Probabilistic Primality Test, therefore the error can be arbitrarily reduced.

Examples

x <- as.vli("89235489145293876129784691")
Pi(x)