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Playing with primes in excel I came to a pattern that I do not understand and I would like to know more about it.

Example:

  |Prime numbers | Multiplies | Subtraction of  | Difference of 
  |              |            | multiplies      | subtraction
--|---------------------------------------------------------------
p1|2             |            |                 |
p2|3             |  3x2 =  6  |                 |
p3|5             |  5x3 = 15  | 15-6  =  9      |
p4|7             |  7x5 = 35  | 35-15 = 20      | 20-9  = 11
p5|11            | 11x7 = 77  | 77-35 = 42      | 42-20 = 22
p6|13            | 13x11=143  | 143-77= 66      | 66-42 = 24
p7|17            | 17x13=221  | 221-143=78      | 78-66 = 12
p8|19            | 19x17=323  | 323-221=102     | 102-78= 24

Simple histogram showed that for first 65535 primes, number 24 is most frequent 'Difference of subtraction of consecutive prime multiplication' (number in 4th column of example table, for prime numbers 13 and 19 is that number equal 24).
Filtering only rows, where this value is 24 we get only twin prime numbers (second from twin pair) or primes with distance 10 (again second number from pair).
At the end if we filter only rows with twin primes, we receive only list of primes that end with 9.

Q1: Does this pattern remain with higher numbers too?
Q2: Is number 24 most frequent 'Difference of subtraction of consecutive prime multiplication' also if we take more primes?
Q3: That would mean that some prime numbers occur in let's say some groups?
Q4: Is there any study or publication which talks about this closer?

Radouxca
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  • How can the difference between twin primes be $24$??? How can the difference between primes with distance $10$ be $24$??? I suggest you edit your question to make it a little more intuitive for reading. – barak manos Mar 21 '16 at 09:30
  • I've edited question to explain little bit more what I mean. – Radouxca Mar 21 '16 at 10:03
  • Note that -for instance- $\small 35-15 = (7-3) \cdot 5 = 4 \cdot 5 $ , $\small 77-35 = (11-5) \cdot 7 = 6 \cdot 7 $ and then the difference is expressible by all primes previously involved. Now differences between odd primes are even, and primegaps are often divisible by 6. Analyzing this in more detail is - I think - promising to explain the frequencies pattern. – Gottfried Helms Mar 21 '16 at 18:13
  • I've added to each line prime number identification p1, p2, etc. – Radouxca Mar 22 '16 at 07:12
  • Now, inspired by previous comment from @Gottfried Helms, the number in fourth column of a table can be written as: $p_6p_5-2p_5p_4+p_4p_3$ . Sorry for putting two comments, but there are rules I'm not familiar with. I just wanted to put new line and comment was sent. Then I wanted to edit it, but there is 5 minute edit limit for comment... – Radouxca Mar 22 '16 at 07:25
  • Concerning the broken comment: no problem. When I have messes with the "comment" I edit /copy text into clipboard/then into notepad/correct/ then delete old comment and write a new one. (Shit happens...) – Gottfried Helms Mar 22 '16 at 08:11
  • I've noticed that there are always (or very often, I do not know it exactly) on place where this 24 appears, prime numbers with gaps 2-4-2 or 10-2-10. For these gaps equation $p_6p_5-2p_5p_4+p_4p_3=24$ has a solution. So the question changes slightly and now I could ask, are there other known "groups" of prime numbers similar to this? Does these groups appear infinite? Is there any theory behind such prime numbers grouping? – Radouxca Mar 22 '16 at 22:09
  • Maybe just an indirect reflection of the distance between consecutive primes. – daniel Mar 24 '16 at 07:25
  • Realise that when you multiply two twin prime numbers together $t_1$ and $t_2$, this is equal to $$\bigg(\frac{t_1 + t_2}{2} + 1\bigg)\bigg(\frac{t_1 + t_2}{2} - 1\bigg) = \bigg(\frac{t_1 + t_2}{2}\bigg)^2 - 1$$ Notice also that $24 = 5^2 - 1$... So perhaps this could explain some kind of relation? – Mr Pie Jan 15 '18 at 01:18

1 Answers1

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After all I can note that question is based on set of 4 consecutive primes and their distances and so these links are partially answers to questions Q1, Q3 and Q4:
https://en.wikipedia.org/wiki/Prime_k-tuple
https://en.wikipedia.org/wiki/Prime_quadruplet

While the questions were not primarily about prime gaps, but about gaps between 'differences of consecutive prime products', I cannot say that mentioned links answer questions fully.

Equation $p_6p_5−2p_5p_4+p_4p_3=24$ has one of its solution prime quadruplet {p, p+2, p+6, p+8}. There is also second solution {p, p+10, p+2, p+10}.
Next question would be, if there are other solutions of this equation, which are also consecutive prime numbers and if yes, what are they.

Radouxca
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