Is there any formula for finding nth prime number?

While there are no practically efficient or simple formulas for finding the nth prime number, several theoretical and complex formulations do exist. Here are some key points:

Encoding Primes

Formulas like those proposed by Sierpinski and Hardy and Wright involve encoding the list of primes into a constant, but these are highly inefficient and of recreational use only. For example, Sierpinski suggested a constant (A) that, when used with the floor function, could generate prime numbers, but this method is not practical for finding the nth prime without first knowing the primes1.

Counting Primes

Methods using Wilson's Theorem or other counting arguments, such as those by Willans, Minác, and Hardy and Wright, provide complex summations to count the number of primes up to a given number ((\pi(n))). However, these are also inefficient for direct computation of the nth prime1.

Explicit Formulas

There are some explicit but highly complex formulas introduced by mathematicians like L. Veshenevskiy and others. These formulas often involve recursive or iterative processes and use various arithmetic functions. For instance, Veshenevskiy's formula from 1962 involves a series of nested expressions that ultimately yield the nth prime number, but these are not practical for computation due to their complexity3.

Practical Approaches

In practice, finding the nth prime number is more efficiently done using algorithms like the Sieve of Eratosthenes, which generates all primes up to a given limit. This method is widely used in programming to find the nth prime number by generating a list of primes and then selecting the nth element from this list24.