dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day! A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

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

dCode retains ownership of the "Prime Counting Function" source code. Except explicit open source licence (indicated Creative Commons / free), the "Prime Counting Function" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or the "Prime Counting Function" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Prime Counting Function" are not public, same for offline use on PC, tablet, iPhone or Android !
The copy-paste of the page "Prime Counting Function" or any of its results, is allowed as long as you cite the online source https://www.dcode.fr/prime-number-pi-count
Reminder : dCode is free to use.