# 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)

• This question is relevant. The answer of Dan Petersen, in particular, contains a link to a nice article by Underwood Dudley that contains some history. – Will Orrick Mar 22 '15 at 18:28
• If he was "rewarded"? – anomaly Mar 22 '15 at 20:33
• The formula here doesn't actually give any insight into computing the $n$th prime efficiently, all it does is encode (obfuscate?) the process of trial division, or something related, into an unreadable mathematical expression. The presence of floor operations and sums (which are a cute but in this context uninteresting way to emulate conditionals and loops) give it away. – Thomas Mar 23 '15 at 7:33
• If "he" was rewarded? – Eli Rose Mar 23 '15 at 9:29
• It is not elementary functions. – Anixx Mar 23 '15 at 13:40

## 2 Answers

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

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:

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

• This is more like a computer program rather than a closed-form formula... – user541686 Mar 23 '15 at 9:44
• @Mehrdad: That is precisely what I was implying at the beginning (top part) of the answer. BTW, I'm not sure that there even exists a definition that accurately distinguishes between a computer program and a closed-form formula (i.e., every computer program can be converted into a formula and vice-versa). – barak manos Mar 23 '15 at 9:46
• I mean, it would be a lot cooler if you could do the same thing in terms of mathematically "nice" functions (e.g. perhaps smoother... no floors/ceilings/absolute values/etc.). And yeah, I don't think there's a rigorous distinction between the two... it's an "I know it when I see it" kind of thing. – user541686 Mar 23 '15 at 9:54
• @Mehrdad: Yep. And in addition to that, unless you can provide a mathematical definition of the term "nice", it remains subjected to personal perspective. – barak manos Mar 23 '15 at 9:58
• How about nice = elementary? – user541686 Mar 23 '15 at 10:00