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Tool for counting prime numbers via the prime-counting function denoted pi(n) which counts the prime numbers less than or equal to a real number n.

Answers to Questions

What is the counting prime function? (Definition)

The counting prime numbers function, called \( \pi(n) \), aims to count the prime numbers less than or equal to a number \( n \).

How to calculate pi(n)?

For small numbers, the easiest method to count all the first primes less than \( n \) is to use the Eratosthenes sieve to quickly list prime numbers.

Example: \( \pi(100) = 25 \) so there are 25 prime numbers less than 100.

How to calculate an approximation of pi(n)?

The value of pi(n) approaches \( n / \ln(n) \) when \( n \) is very big:

$$ \lim_{ n \to + \infty } \pi(n) = \frac{ n }{ \ln(n) } $$

This formula is also called the prime number theorem.

What is pi(n) for?

The calculation of pi(n) allow to locate a prime number with respect to another, knowing its rank in the list of prime numbers.

If pi(a) < pi(b) then a < b.

How to get an estimation of the nth prime number?

A consequence of the prime number theorem is that the nth prime number \( p_n \) is close to \( n \ln(n) \) (and closer when \( n \) is very large) $$ p_n \sim n \ln (n) $$

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