# Consecutive prime numbers multiplication pattern

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?

• 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. Mar 21, 2016 at 9:30
• I've edited question to explain little bit more what I mean. Mar 21, 2016 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. Mar 21, 2016 at 18:13
• I've added to each line prime number identification p1, p2, etc. Mar 22, 2016 at 7:12
• 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... Mar 22, 2016 at 7:25

## 1 Answer

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.