# Mills' Theorem/Lemma 3

## Lemma for Mills' Theorem

Let:

$\N$ denote the set of all natural numbers
$\Bbb P$ denote the set of all prime numbers.

Let $p_n$ be the $n$th prime number.

$p_{n + 1} - p_n < K {p_n}^{5 / 8}$

where $K$ is an unknown but fixed positive integer.

Let $P_0 > K^8$ be a prime number.

By Lemma 1, there exists an infinite sequence of primes:

$P_0, P_1, P_2, \ldots$

such that:

$\forall n \in \N_{>0}: {P_n}^3 < P_{n + 1} < \paren {P_n + 1}^3 - 1$

Let us define the mapping $v: \N \to \Bbb P$ as:

$\forall n \in \N: \map v n = \paren {P_n + 1}^{3^{-n} }$

Then we have that:

$\forall n \in \N_{>0}: \map v {n + 1} < \map v n$

## Proof

 $\ds \map v {n + 1}$ $=$ $\ds \paren {P_{n + 1} + 1}^{3^{-\paren {n + 1} } }$ $\ds$ $<$ $\ds \paren {\paren {\paren {P_n + 1}^3 - 1} + 1}^{3^{-n - 1} }$ because $P_{n + 1} < \paren {P_n + 1}^3 - 1$ $\ds$ $=$ $\ds \paren {\paren {P_n + 1}^3}^{3^{-n - 1} }$ $\ds$ $=$ $\ds \paren {P_n + 1}^ {3^{-n} }$ $\ds$ $=$ $\ds \map v n$

$\blacksquare$