Tackling Andrica's conjecture. Part 3

Here is an interesting, but slightly detached from the previous two articles, result. Let's look at the following two sequences: $$a_{n}=\sqrt{p_{n+1}} - \sqrt{p_{n}}$$ $$b_{n}=\ln\left ( \frac{1+\sqrt{p_{n+1}}}{1+\sqrt{p_{n}}} \right )^{1+\sqrt{p_{n+1}}}$$

Here is a short Python code to visualise the sequences.

And here is how both sequences look like ($a_{n}$ the first and $b_{n}$ the second): Quite asymptotic, aren't they? Indeed they are ...

Lemma 3. $$\sqrt{p_{n+1}} - \sqrt{p_{n}} \leq \ln\left ( \frac{1+\sqrt{p_{n+1}}}{1+\sqrt{p_{n}}} \right )^{1+\sqrt{p_{n+1}}} \leq \left ( \frac{1+\sqrt{p_{n+1}}}{1+\sqrt{p_{n}}} \right )\cdot \left ( \sqrt{p_{n+1}} - \sqrt{p_{n}} \right )$$

Let's look at this function $f_{6}(x)=\frac{\sqrt{p_{n+1}}}{1+x\cdot \sqrt{p_{n+1}}}$. Obviously, ${\ln\left ( 1+x\cdot \sqrt{p_{n+1}} \right )}'=f_{6}(x)$. As a result $$\int\limits_{\sqrt{\frac{p_{n}}{p_{n+1}}}}^{1} f_{6}\left ( x \right )dx = \ln\left ( 1+x\cdot \sqrt{p_{n+1}} \right )|_{\sqrt{\frac{p_{n}}{p_{n+1}}}}^{1}=\ln\left ( \frac{1+\sqrt{p_{n+1}}}{1+\sqrt{p_{n}}} \right )$$

According to Mean Value Theorem, $\exists \mu \in \left (\sqrt{\frac{p_{n}}{p_{n+1}}} ,1 \right )$ such that: $$\int\limits_{\sqrt{\frac{p_{n}}{p_{n+1}}}}^{1} f_{6}\left ( x \right )dx = f_{6}\left ( \mu \right )\cdot \left ( 1- \sqrt{\frac{p_{n}}{p_{n+1}}} \right )$$

Putting all together: $$\ln\left ( \frac{1+\sqrt{p_{n+1}}}{1+\sqrt{p_{n}}} \right ) = \frac{\sqrt{p_{n+1}}-\sqrt{p_{n}}}{1+\mu \cdot \sqrt{p_{n+1}}}$$

Because $$\sqrt{\frac{p_{n}}{p_{n+1}}}< \mu < 1 \Rightarrow 1+\sqrt{p_{n}}< 1+\mu \cdot \sqrt{p_{n+1}} < 1+\sqrt{p_{n+1}} $$

And we get $$\frac{\sqrt{p_{n+1}}-\sqrt{p_{n}}}{1+\sqrt{p_{n+1}}}\leq \ln\left ( \frac{1+\sqrt{p_{n+1}}}{1+\sqrt{p_{n}}} \right )\leq \frac{\sqrt{p_{n+1}}-\sqrt{p_{n}}}{1+\sqrt{p_{n}}}$$ which proves this lemma.

Noting $\Delta_{n}=\sqrt{p_{n+1}}-\sqrt{p_{n}}$, this becomes: $$\frac{\Delta_{n}}{1+\sqrt{p_{n+1}}}\leq \ln\left ( 1 + \frac{\Delta_{n}}{1+\sqrt{p_{n}}} \right )\leq \frac{\Delta_{n}}{1+\sqrt{p_{n}}}$$ or $$\frac{1+\sqrt{p_{n}}}{1+\sqrt{p_{n+1}}}\leq \ln\left ( 1 + \frac{\Delta_{n}}{1+\sqrt{p_{n}}} \right )^{\frac{1+\sqrt{p_{n}}}{\Delta_{n}}}\leq 1$$

Is this result of any use? I don't know yet, but it looks like: $$\left ( 1 + \frac{\Delta_{n}}{1+\sqrt{p_{n}}} \right )^{\frac{1+\sqrt{p_{n}}}{\Delta_{n}}} \rightarrow e, n \to \infty$$ using this result.