I was looking at A Conjecture About Prime Numbers and I went into the topic a little bit further.
Say you are given the equation: $$\mathcal{N} = \{p_n - p_{n + 1} - p_{n + 2} + p_{n + 3} : p_n = \text{$n^{th}$ prime number}\}$$ It seems like there are many solutions where $\mathcal{N} = 0$: $$\begin{align} 0 &= 5 - 7 - 11 + 13 &= p_3 - p_4 - p_5 + p_6 \\ &= 7 - 11 - 13 + 17 &= p_4 - p_5 - p_6 + p_7 \\ &= 11 - 13 - 17 + 19 &= p_5 - p_6 - p_7 + p_8 \\ &= 13 - 17 - 19 + 23 &= p_6 - p_7 - p_8 + p_9 \end{align}$$ And then $0 \neq 17 - 19 - 23 + 29 \ \lor \ 19 - 23 - 29 + 31$ but then: $$0 = 23 - 29 - 31 + 37 = p_9 - p_{10} - p_{11} + p_{12}$$ It quickly becomes clear that: $$0 = p_n - p_{n + 1} - p_{n + 2} + p_{n + 3} \iff p_{n + 1} - p_n = p_{n + 3} - p_{n + 2}$$
Question:
Given the expression: $$p_n - p_{n + 1} - p_{n + 2} + p_{n + 3} : p_n = \text{$n^{th}$ prime number}$$ What is the ratio between answers equal to $0$ and answers not equal to $0$, if there is a ratio? If there isn't a ratio, how can we prove so?
I have no idea where to begin, so I would mostly appreciate full answers and not hints, unless it is a pretty strong hint.
Thank you in advance.