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:

import math
import matplotlib.pyplot as plt
primes = []
def isPrime(n):
l = int(math.sqrt(n)) + 1
for i in xrange(2,l):
if (n % i) == 0:
return False
return True
def calculateLog(sqrt_p1, sqrt_p2):
sqrt_1_p2 = 1.0 + sqrt_p2
r = math.log(sqrt_1_p2/(1.0 + sqrt_p1))
return r * sqrt_1_p2
N = 1500000
print "populate primes ..."
for i in xrange(2, N):
if isPrime(i):
primes.append(i);
sqrt_diff = [] # sqrt diffs
diff = [] # simple diffs
log_calcs = [] # log calcs
x = []
for i in xrange(1, len(primes)):
sqrt_p2 = math.sqrt(primes[i])
sqrt_p1 = math.sqrt(primes[i-1])
sqrt_diff.append(sqrt_p2 - sqrt_p1)
diff.append(primes[i] - primes[i-1])
log_calcs.append(calculateLog(sqrt_p1, sqrt_p2))
x.append(i)
for i in xrange(len(sqrt_diff)):
print sqrt_diff[i]," = s(",primes[i+1],") - s(",primes[i],")"
plt.subplot(311)
plt.plot(x, sqrt_diff)
plt.subplot(312)
plt.plot(x, log_calcs)
plt.subplot(313)
plt.hist(diff, 1000)
plt.show()

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 $$\int_{\sqrt{\frac{p_{n}}{p_{n+1}}}}^{1} f_{6}\left ( x \right )dx = ln\left ( 1+x\cdot \sqrt{p_{n+1}} \right )|_{\sqrt{\frac{p_{n}}{p_{n+1}}}}^{1}=ln\left ( \frac{1+\sqrt{p_{n+1}}}{1+\sqrt{p_{n}}} \right )$$

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

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.