What is the formula for finding the nth prime no?

Answer 1

There isn't an exact formula. I cannot tell you the 5th prime other than by trying every number until I find the 5th prime.

What you can do is sieve for primes. noting that 0, 1 are neither prime nor composite, make a list starting from 2. Follow these rules: Every number not crossed out is a Prime number. Once a prime number is found, count that number forward, and cross out that number and every other number in the list. E.g. 2 is the first prime, so, cross out every other number in the list (as these are all multiples of 2). 3 is the next prime, so cross out every 3rd number. 4 has been crossed out (due to 2), and 5 is the next prime, so cross out every 5th number. 6 has been crossed out (due to 2 and 3), so skip it, and 7 is not crossed out, so that is the next prime.

this process is infinitely repeatable but at the same time one could never complete this process because there are infinite numbers. As such, this method really only works for a small set of numbers. By hand, maybe all primes 2-2000, by computer, maybe 2-2,000,000,000 or more.

See Wikipedia's article on the Prime Counting Function. Its approximation allows one to target the expected area of the nth prime as the nth prime will occur near 1.02*n/ln(n). This approximation says the 1107th prime should occur around 10000. It's actual location is at 8887. Similarly, we get estimate, actual for the 8859th prime as 100K vs 91673.