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I'm writing a program that finds the nth prime number, so if n was 10, the answer would be 29. I need to find an upper bound to search through though so if n was 10, the upper bound would be 30 (or any number before the next prime number which is 31).

Any ideas?

James Mitchell
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  • It seems a lot easier to sieve (I have no idea if there exist any good bounds). – pjs36 Oct 12 '15 at 17:57
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    Wikipedia: Approximations for the nth prime number – Martin R Oct 12 '15 at 18:01
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    Also see the questions:

    http://math.stackexchange.com/questions/507178/most-efficient-algorithm-for-nth-prime-deterministic-and-probabilistic and http://math.stackexchange.com/questions/1411346/how-do-i-findisolate-the-n-th-prime-number

    As they give ideas for how to efficiently implement your actual program (find nth prime) for relatively large inputs.

     – DanaJ Oct 12 '15 at 22:42

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The number of primes below a given number, $\pi(x)$ can be estimated by:

$\pi(x) \approx \frac{x}{\ln(x)}$.

If you can find a "good" way to invert that function (and add maybe 20% uncertainty in there for small numbers), you can find the estimated upper bound.

costrom
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    The $r = \pi \ln \pi$ and $s = \pi (\ln \pi + \ln \ln \pi)$ are first and second-order approximations for retrieving $x$ from costrom's formula. For $\pi = 10$, these yield $r=23$ and $s=31$. – user3294068 Oct 12 '15 at 18:20