3

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.

Mr Pie
  • 9,459
  • 1
    If you define prime gaps as $g_k = p_{k+1} - p_k$, then you're asking for how often $g_n = g_{n+2}$. Due to how so very little is generally known regarding how prime gaps relate to each other near by, I'm quite certain there's no particular ratio that anybody can reasonably provide for you. Also, FYI, a similar question was asked about $3$ weeks ago at Are there infinite many prime-"tuples", for which $p_{n+1}+p_{n+2}−{p_n}=p_{n+3}$. – John Omielan Feb 18 '19 at 04:23
  • @JohnOmielan thanks for letting me know, of your comment and of the question as well! :) – Mr Pie Feb 24 '19 at 04:22
  • 1
    You are welcome. Perhaps some day we will be able to learn enough about how primes behave overall in alternate prime gaps to give you are reasonable answer regarding the ratio of the values you are asking about. – John Omielan Feb 24 '19 at 04:32
  • @JohnOmielan it might sound silly, but one day... I will. No matter how small, I will find SOMETHING (...if I haven't already, hehe). I will start from there, and just keep going and getting bigger. I have a dream! (Yeah, so what if Martin Luther King Jr knew nothing about math, I can still use his quote.) – Mr Pie Feb 24 '19 at 10:13
  • 1
    I conjecture the ratio is zero. The function ratfun(n)={s=0;for(i=1,n,if(prime(i)-prime(i+1)-prime(i+2)+prime(i+3)==0,s=s+1));print(s/n);} tends to decrease as n increases. – ə̷̶̸͇̘̜́̍͗̂̄︣͟ Jul 19 '19 at 09:28

0 Answers0