TITLE: Who discovered the first explicit formula for the n-th prime?
QUESTION [14 upvotes]: 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)

REPLY [19 votes]: 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:
$$P_n=\sum\limits_{k=n}^{n^2+1}{k}\cdot{F_k}\cdot\left(1-\left\lceil\frac{(G_k-n)^2}{(G_k+n)^2}\right\rceil\right)$$