Who discovered the first explicit formula for the n-th prime? I just found out on Wolfram that there is a formula for the n-th prime in terms of elementary functions. I wonder who found it and if he was rewarded for this. The formula (here) is: 

Also shown at 
http://www.wolframalpha.com/input/?i=prime(n)
 A: Here is something that I have established a long time ago. It doesn't answer your question directly, but rather illustrates the fact that one can easily establish such formula (the one below is given by $P_n$). The real challenge is to establish a prime-formula which is not "computationally worthless".

Is $n$ prime:
$$F_n=\left\lfloor\frac{\left(\sum\limits_{k=2}^{n-1}\left\lceil\frac{{n}\bmod{k}}{n}\right\rceil\right)+2\cdot\left\lceil\frac{n-1}{n}\right\rceil}{n}\right\rfloor$$

How many primes until $n$:
$$G_n=\sum\limits_{k=2}^{n}F_k$$

What is the $n$th prime number:
$$P_n=\sum\limits_{k=n}^{n^2+1}{k}\cdot{F_k}\cdot\left(1-\left\lceil\frac{(G_k-n)^2}{(G_k+n)^2}\right\rceil\right)$$
A: There are many formulas for the $n$th prime.
The only really useful ones are in the form of fast computer programs
(yes, those are formulas too, just not what you would normally consider a
"closed-form" formula of elementary functions).
Consider the formula in question here,

Observe that in order to find the $n$th prime, this formula requires you to
compute $2^n$ terms of the sum. Hence as an algorithm it is of order $\Omega(2^n)$,
which is pretty bad.
In addition to the question The myth of no prime formula? already cited in a comment,
see the following


*

*What would be the immediate implications of a formula for prime numbers

*Any formula for the exact number of primes below a given bound

*Can insight be derived from direct formulae for prime number functions

*Formula for the nth prime number: discovered?
