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
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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) $$ p_n \sim n \ln (n) $$
Source code
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