Upper Bounds for Prime Numbers

From ProofWiki
Jump to navigation Jump to search


Let $p: \N \to \N$ be the prime enumeration function.

Then $\forall n \in \N$, the value of $\map p n$ is bounded above.

In particular:

Result 1

$\forall n \in \N: \map p n \le 2^{2^{n - 1} }$

Result 2

$\forall n \in \N: \map p n \le \paren {p \paren {n - 1} }^{n - 1} + 1$

Result 3

$\forall n \in \N_{>1}: p \left({n}\right) < 2^n$


In the first two cases it can be seen that the limit found is wildly extravagantly large.

However, they are results that are easily established, and they have their uses.