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December 12th, 2016, 05:40 AM  #11 
Senior Member Joined: Mar 2015 From: England Posts: 199 Thanks: 5 
I found some new things. 1st row number = n Last row number = n+(n*2+1) / Row position of Current P + 1 (if shift left) or + 2 finds next row position of Current P on next row. / Row position of Current P + 2 * x = row position of Next P on current row. Here is the list of x starting from the first 0 http://oeis.org/search?q=0%2C2%2C2%2...lish&go=Search Last edited by HawkI; December 12th, 2016 at 05:42 AM. 
December 12th, 2016, 08:12 AM  #12 
Senior Member Joined: Mar 2015 From: England Posts: 199 Thanks: 5 
Just realised the "Row position of Current P + 2 * x = row position of Next P on current row" is basically Current P + 2*n to next Prime so. For example 3 + 2 = 5 5 + 2 = 7 etc 
December 15th, 2016, 04:10 PM  #13 
Senior Member Joined: Mar 2015 From: England Posts: 199 Thanks: 5 
I was looking into Pascal's triangle the other day and how if there's a Prime number after the number 1 on a row it can divide all the other numbers in that row. This inspired a new way of looking into this "Relative Fraction Trapezium" and what I found was interesting. Not only is the next Prime number guaranteed to be in the finite row, but also now for the first time, it some times tells you what the next Prime number is through the use of finding the Average of Irrational rotations and or adding up the Integral values. Edit: A neater copy https://drive.google.com/file/d/0B7M...ew?usp=sharing Last edited by HawkI; December 15th, 2016 at 04:15 PM. 
December 17th, 2016, 01:05 PM  #14 
Senior Member Joined: Mar 2015 From: England Posts: 199 Thanks: 5 
I just wanted to let you know that I am very close to finding how to do this Prime pattern automatically now and that if you want to be mentioned in this discovery (Rediscovery) then your time is running out. I mention this because just now I realised I can manually find the value of "x" which divided by Pronic equals to round down 1 / Infinite round down. And then from there + OR  by n to get to the Prime in negative form. 
March 2nd, 2017, 09:29 AM  #15 
Senior Member Joined: Mar 2015 From: England Posts: 199 Thanks: 5 
On and off I have been looking at this and here are some new things I have found You can consider each row of the trapezium as a set. You can always find an intersection between the current set and the set below. Find it to the right of the current Prime and the next Prime in the next set is directly below Find it to the left of the current Prime and the next Prime in the next set is directly south west diagonal Surely by now I have proved in some shape or form that Primes are a recurrence relation? Last edited by HawkI; March 2nd, 2017 at 09:43 AM. 
April 16th, 2017, 11:58 AM  #16 
Senior Member Joined: Mar 2015 From: England Posts: 199 Thanks: 5 
I have found some new things, it's been staring at me this whole time. Prime, Prime / n = x.y modulo congruence n = a, n Next Prime = Colum Set First Positive n  n * (a + 1) * 2, n + 1 (n + 1) + (n + 1)squared = Pronic n + 4 = Colum Space Colum Start + ((n + 1) * Colum Space) = Colum Set Colum Start = Pronic + (n * z) + (((n+1) * Colum Space)) + (n + 1) >= Pronic 
April 16th, 2017, 12:07 PM  #17 
Senior Member Joined: Mar 2015 From: England Posts: 199 Thanks: 5 
This doesn't work when the next prime / n is a + 1, but I thought that this was good enough to share anyway. EDIT: 11, 8, 5, 2, 1, 4, 7, 10 though it goes up in 3s I noticed that 2 * 2 = 4 and that 12  4 = 8 2 starts at 8 + 1 * 3 = 5 the Prime And as a proof check the Primes add the first positive always go up in multiples, you can refer back to the diagrams I have on Google drive to see this for your self. You will find this on the Sheet 2 button. EDIT: Also 1 * 2 = 2 12  2 = 10 from 8 to 10 is 7 4 + 3 = 7 2 + 1 = 3 5 is the 3rd Prime Last edited by HawkI; April 16th, 2017 at 12:42 PM. Reason: Visual diagram of my latest finding 
April 20th, 2017, 11:14 AM  #18 
Senior Member Joined: Mar 2015 From: England Posts: 199 Thanks: 5 
After looking into this on and off for maybe a year and a half, thinking this is a possible pattern for Prime numbers, spurred on with the use of Pronics and the similarities with this and Sacks Spiral, I must concede and say that after all this time, although this looks like a pattern... it is just a cumbersome way to add numbers. This is not a pattern. EDIT: So I just looked back on my first thread about this and I would say I spent around nine months on this. Last edited by HawkI; April 20th, 2017 at 11:48 AM. Reason: TIme correction 

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