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.
Prime Counting Function - dCode
Tag(s) : Arithmetics
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Prime Counting Function
Answers to Questions (FAQ)
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 $ as 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 $ tends to infinity (ie. $ n / \ln(n) $ is a good approximation of $ pi(n) $ when $ n $ is very large)
$$ \pi(n)\ \underset{ n \to \infty }{ \sim } \frac{n}{\ln(n)} $$
This formula is also called the prime number theorem.
What is pi(n) for?
The calculation of pi(n) allows 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) $$ p_n \underset{ n \to \infty }{ \sim } n \ln (n) $$
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