# Upper Bounds for Prime Numbers

## Theorem

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

Then $\forall n \in \N$, the value of $p \left({n}\right)$ is bounded above.

In particular:

### Result 1

$\forall n \in \N: p \left({n}\right) \le 2^{2^{n-1}}$

### Result 2

$\forall n \in \N: p \left({n}\right) \le \left({p \left({n - 1}\right)}\right)^{n-1} + 1$

### Result 3

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

## Note

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.