The following limit (after analyzing 3 successive primenumbers) was found:
$$\lim_{n \rightarrow \infty}\frac{3g_{n}^2}{p_{n}}=0$$
$$g_{n}\ll \sqrt{\frac{p_{n}}{3}}$$
Thanks to comments I could trace back that it looks like: Oppermann's conjecture and a consequence of Lindelöf hypothesis.
The method is explained below. I was hoping someone could explain more about the found pattern (have identical methods been used before?). Is the method is valid and what would it require to be an proof?
Method.
Two functions are defined: $\varepsilon_1$ and $\varepsilon_2$. Function $\varepsilon_1$ is based upon prime triangles created from 3 following prime numbers. Function $\varepsilon_2$ is based upon the error with regard to a balanced prime number. More information: Error Prime Prediction With Prime Triangles (Q: growth and symmetry).
$$\varepsilon_{1}(n)=\frac{1}{2}{p}_{n-2}-p_{n}+\sqrt{-\frac{3}{4}{p}_{n-2}^{\:2}+{p}_{n-1}^{\:2}}$$
$$\varepsilon_{2}(n)=2{p}_{n-1}-{p}_{n-2}-{p}_{n}$$
$$\Delta\varepsilon(n)=\varepsilon_{1}(n)-\varepsilon_{2}(n)$$
$$\lim_{n \rightarrow \infty}\Delta\varepsilon(n)=0$$
The difference between both functions $\Delta\varepsilon$ converges to $0$ for $n\rightarrow \infty$ [Limit Wolfram Alpha].
Function analysis has been done on $\Delta\varepsilon$ for the first 14.000.000 primes. It is observed that $1/\Delta\varepsilon$ correlates with $p_{n}$. Appearing (no proof) straight lines occur for the prime gaps (see graph). Explanation: the error distribution $\varepsilon$ in [SE: Prime Triangles] shows hyperbolic features.
The slopes and intercept for each prime gap can be calculated. The slopes appear to correlate linear on log scale. The following formula can then be found for the slope:
$$\log(1/\Delta\varepsilon)=slope \cdot\log(p_{n})+intercept$$
$$intercept\approx-2.0021\cdot\log(g_{n-2})-1.0893$$
$$slope \rightarrow 1$$
And so $\Delta\varepsilon^\prime(n)$ (prime) can be calculated with:
$$\Delta\varepsilon^\prime(n)=-\frac{ag_{(n-2)}^b}{p_{n}}\approx-\frac{3g_{(n-2)}^2}{p_{n}}$$
$\Delta\varepsilon^\prime$ and $\Delta\varepsilon$ both converge to $0$. $\Delta\varepsilon^\prime$ converges slower than $\Delta\varepsilon$. Analysis shows that $a=3$ and $b=2$ have the best fit with slower convergence [Wolfram Analysis]. Also learned from comments: $g_{n-2}\sim g_{n}:$
$$\lim_{n \rightarrow \infty}\frac{3g_{n}^2}{p_{n}}=0$$
$$g_{n}\ll \sqrt{\frac{p_{n}}{3}}$$
Error difference $\Delta\varepsilon^\prime-\Delta\varepsilon$ is plotted below (residual error). Patterns appear but become diffuse, no new patterns where found. Graph is plotted for positive and negative errors. Understanding this group $\Delta\varepsilon^\prime-\Delta\varepsilon$ possibly improves the prime gap bound earlier found.
Questions:
I was hoping someone could explain more about the found pattern (have identical methods been used before?). Is the method is valid and what would it require to be an proof?
Any feedback is welcome.
Best Answer
Analytical approach.
The question shows a numerical method to derive the prime gap limit. Below a analytical method is used to come to the same prime gap limit:
$$\varepsilon_{1}(n)=\frac{1}{2}{p}_{n-2}-p_{n}+\sqrt{-\frac{3}{4}{p}_{n-2}^{\:2}+{p}_{n-1}^{\:2}}$$ $$\varepsilon_{2}(n)=2{p}_{n-1}-{p}_{n-2}-{p}_{n}$$
$$\Delta\varepsilon(n)=\varepsilon_{1}(n)-\varepsilon_{2}(n)$$
$$\Delta\varepsilon(n)=1.5p-2(p+g)+\sqrt{-0.75p^{2}+(p+g)^{2}}$$
According Wolfram Alpha this expression can be written as the following series Series Wolfram Alpha:
$$\Delta\varepsilon^{\prime}(n)=-\frac{3g^{2}}{p}+\frac{12g^{3}}{p^{2}}-\frac{57g^{4}}{p^{3}}+\frac{300g^{5}}{p^{4}}-\frac{1686g^{6}}{p^{5}}+\mathcal{O}\left( \frac{1}{p^{6}}\right)$$
So also with this method the same limit is found like the question:
$$\lim_{n \rightarrow \infty}\Delta\varepsilon(n)=0$$ $$\lim_{n \rightarrow \infty}-\frac{3g_{n}^{2}}{p_{n}}=0$$
This is no proof but the results are the same as the numerical method in original question. It appears that $\Delta\varepsilon$ between: $\varepsilon_1$: error triangles with prime length and $\varepsilon_2$ error with respect to balanced prime might explain prime gap patterns.
Below a graph of the error remainder: $\Delta\varepsilon-\Delta\varepsilon^{\prime}<0$ (basically: $\mathcal{O)}$. Note that no positive error remainder are observed like for numerical method (original question) did. Observations Error remainder: