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The sequence of prime numbers is the set of prime numbers in their natural order (that is, $2, 3, 5, 7, 11, 13, 17,...$). The German wikipedia entry on sequences states the following:

Given $i$, there is no other way to name the $i$-th prime number than to calculate the whole sequence of prime numbers from the first to the $(i-1)$-th element.

Is this provably true and if so, how can it be proven?

vauge
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  • As far as I know, this is not provably true. The problem is rather that there is no known formula (up to now), which allows us to compute the $i$-th prime in an efficient way. See e.g. here: https://en.wikipedia.org/wiki/Formula_for_primes. – russoo Feb 13 '15 at 15:15
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    I believe that if a formula exists, it's probably so complicated you're better off just sieving up to $i$. – Mr. Brooks Feb 13 '15 at 22:14
  • you in fact don't need sieving up to i. You need sieving from (i-1) and up. – user25406 Feb 14 '15 at 03:53

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That doesn't seem like it can be true, as we have implementations that are much faster than generating primes. See the answers to this question: Most efficient algorithm for nth prime, deterministic and probabilistic?.

Simplistically, you use an n-th prime approximation to get you very close, run a fast prime count algorithm to give you the exact answer at the approximation, then sieve and count the difference.

Edit: To relate to the wikipedia entry, not only do we have another way, but it is a much faster way. The statement is false. We don't have a useful explicit formula, which is probably more what the author meant to convey.

Community
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DanaJ
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