Here we address the following frequently asked question.
Is there a formula for producing a specific prime number? Let
me give you an example. The formula is given the number 52. In return,
the formula produces 239, the fifty-second prime number (I think).
Yes, there are many such formulas--but they are of recreational use only
because they are very inefficient. Most are either ways of encoding the
list of primes or very clever counting arguments. I will give a couple example
below, but for more information start with chapter three "Are There Functions
Defining Prime Numbers?" of Ribenboim's text [Ribenboim95
Method One: Encoding Primes
Let pn be the nth prime. In 1952 Sierpinski
suggested we define a constant A as follows:
A = = 0.02030005000000070...
Then using the floor function [x] (the greatest integer less than
or equal to x) we have
Hardy and Wright [HW79
p345] give a variant of this: Let r be an integer greater than one
and define a constant B as follows:
This type formula would only be of value if the necessary constant could
be found without first finding the primes--this may be possible,
but it seems unlikely.
Method Two: Counting Primes with Wilson's Theorem
First use one of these three methods to define pi(x). Willans (1964) used
pi(n) = (sum from j=2 to n) sin2(pi*(j-1)!2/j)