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Are there any conjectures from which we can infer something about the first occurrences of prime gaps length $n$ and their distribution? I've made an interesting graph of these values to make this problem easier, with gauss's approximate of their size, $\log(p)$ on the $x$-axis and on the $y$-axis the proportional error, i.e. $\frac{g(p)}{\log(p)}$. A prime gap $g(p)$ is $p_{n+1} − p_n$. What on earth is going on with gaps with error from 25 to 30, why are some of these gaps starting so late, does this pattern continue somehow? I needn't say anything about the nature of the large 'line' but does anyone see the vertical 'wave' like pattern with the anomalistic bunch?

Approximately the first 1000 gaps

I hope this question doesn't come across as too wordy or vague but would really appreciate any general comment.

jg mr chapb
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    Could you explain how you created the graph? If we enumerate the primes, then for every pair $(p_n,p_{n+1})$, how do you decide whether to draw a point, and where do you put it? – Chris Culter Aug 01 '15 at 18:37
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    So lets make a set and call it S = {}. A point is drawn if the gap between a prime pair $p_{n+1} - p_n$ produces a value that is not already in this set. Then, we observe gauss's estimate for the prime gap for $p_n$ and put that on the x axis. On the y axis is the proportional error, so the gap divided by log(p) – jg mr chapb Aug 01 '15 at 18:45
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    Also it's for approximately the first 1000 gaps – jg mr chapb Aug 01 '15 at 18:54
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    Hmm, it looks like the large 'line' peters out around $e^{44}\approx 2^{64}$. It is possible that your numbers got truncated to 64 bits somewhere in your calculations? – Chris Culter Aug 01 '15 at 19:06
  • Still not quite sure how you formed the graph, but I would refer you to my answer to this question: https://math.stackexchange.com/questions/2311652/im-trying-to-find-the-longest-consecutive-set-of-composite-numbers/2313788#2313788 – user56983 Aug 13 '17 at 02:32

1 Answers1

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Graham conjecture We can determine a boundary for the first occurance of certain prime gaps if obeying one or more of the two conditions below. If both conditions are obeyed, then we determine that the first occurance of a prime gap is found following the condition which gives the lowest least bound.

condition 1 -symmetric condition for symmetric type gaps. The first symmetric gap of length $2p_{n}$. It is between consecutive primes which are larger than $p_{n+2}^{2}$ and are equal to $k(\#p_{n+1}!) + p_{n}$ and $k(\#p_{n+1}!) - p_{n}$ such that k is the smallest natural number so that they are both larger than $p_{n+2}^{2}$.

Condition 2 - one directional condition for non symmetric gaps.

The first occurance of a non symmetric gap length $p_{n}-1$ is greater than $p_{n-1}^{2}$ and less than $k(\#p_{n-1}!) $where $k$ is the smallest natural number that doesn't produce a contradiction.

Like i said before, and to clarify; the lowest occurance of a gap, if both prerequisites are met - is the lowest boundary given by them. The lowest gap is either symmetric or not, i don't know how to determine this yet, but one if the two conditions will be correct if both prerequisites are satisfied.

Brad Graham
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  • That conjecture is surely false. – quid Aug 02 '15 at 18:05
  • There is one other condition. In the case $p_{n} - 1 = 2p_{m}$ we take the smallest of the two boundary conditions given above. Proof? – Brad Graham Aug 02 '15 at 18:07
  • If anyone finds a counter example let me know, but i have a proof of it being right in my head :) – Brad Graham Aug 02 '15 at 18:14
  • What do you mean exactly by "$#p_{n-1}!$"? – quid Aug 02 '15 at 18:22
  • $p_{1} \times p_{2} \times p_{3} \times ... \times p_{n-1}$ – Brad Graham Aug 02 '15 at 18:25
  • Then, there is a gap of size $22 = 2 \times 11= 2 p_{5}$ after $1129$ while $# p_6 ! $ is over $30000$ and way larger. So the (first) conjecture is false. – quid Aug 02 '15 at 18:38
  • But $1129$ is greater than $19^2$ by the second conjecture. And we need to choose the minimum condition because there are two types of prime gaps that can occur. The first occurance of a prime gap of 22 is not a symmetric gap is the conclusion we draw - it is for the record a one directional one. I havnt looked into determing the type of gap of first occurances yet. – Brad Graham Aug 02 '15 at 19:07
  • I have no idea what you mean to say. Perhaps you could rephrase you conjecture(s) in a more precise way. – quid Aug 02 '15 at 19:10
  • Look at my second comment. – Brad Graham Aug 02 '15 at 19:11
  • Okay. It is still not quite clear. But so there is a prime gap of size $34 = 2 \times 17$ after $1327$. What about that? – quid Aug 02 '15 at 19:13
  • Hmm, what about 36? The first occurance of gap 34 is not symmetric. If its non symmetric, then im guessing the first gap if 34 is actually a gap larger than 34. – Brad Graham Aug 02 '15 at 19:27
  • Condition 2 is plausible (but quite weak). – quid Aug 02 '15 at 19:31
  • For 34, as it is not one less than a prime, we can calculate the boundaries of its first nonsymmetric occurance by using 36 instead. The gap i calculate is a maximum gap length. 34 is bigger then 31 so we can't sub that instead. The first non symmetric gap 34 will occur after $31^{2}$ for example – Brad Graham Aug 02 '15 at 19:55
  • definitely seems heuristic but this result is pretty accurate http://arxiv.org/abs/1102.0481 – jg mr chapb Aug 25 '15 at 11:42
  • For a bit of an explanation take a look at this question http://math.stackexchange.com/questions/1404429/the-longest-sequence-of-numbers-with-a-certain-divisibility-property?noredirect=1#comment2870414_1404429 – Brad Graham Aug 25 '15 at 19:59