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 online "Prime Counting Function" source code. Except explicit open source licence (indicated CC / 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, copy-paste, or API access for "Prime Counting Function" are not public, same for offline use on PC, tablet, iPhone or Android ! Remainder : dCode is free to use.

Need Help ?

Please, check our dCode Discord community for help requests! NB: for encrypted messages, test our automatic cipher identifier!

Questions / Comments

Thanks to your feedback and relevant comments, dCode has developed the best 'Prime Counting Function' tool, so feel free to write! Thank you!

Thanks to your feedback and relevant comments, dCode has developed the best 'Prime Counting Function' tool, so feel free to write! Thank you!