I am wondering why this property exists or if I have discovered a new property of prime numbers?


Let's do a sample here:


2,3,5,7,11,13,17


You obtain the gaps (subtracting the left number from the right number):


1,2,2,4,2,4


You then obtain the gaps of the gaps, following the same procedure:


1,0,2,-2,2


If I follow this type of procedure, why does the sum always tend back to 1? Here is more data illustrating this. Sum up each row, you'll notice the numbers cancel each other out. The **bold** number immediately to the right of the cancelling out is how many steps it took before it cancelled out.


[1] **1**


[0, 2, -2] **3**


2, -2 **2**


2, 2, -4 **3**


4, -2, -2 **3**


2, 2, 0, -4 **4**


4, -2, -2 **3**


4, -2, 2, 2, -4, -2 **6**


2, -2 **2**


2, 10, -10, 2, -4 **5**


8, -8 **2**


4, 0, -2, 2, 0, -4 **6**


8, -8 **2**


2, -2 **2**


10, 0, -8, -2 **4**


2, 2, -4 **3**


8, -4, 0, 0, -4 **5**


4, -2, -2 **3**


8, 4, -10, -2 **4**


2, 10, -8, 4, -8 **5**


2, 2, 2, -2, 0, -2, 2, 2, -4, 4, 2, -8 **12**


8, -8 **2**


4, -2, 2, 2, -4, -2 **6**


2, 8, -4, -4, 4, -4, 2, 6, -10 **9**


16, -12, 4, -4, 0, -4 **6**


4, 4, -4, 0, -4 **5**


4, 0, -2, -2 **4**


10, -2, -8 **3**


2, 2, 0, -4 **4**


10, -8, 2, 2, 2, -2, 2, -2, -2, 0, -2, 4, -2, -2, 4, -4, 10, -4, 2, -10 **20**


8, -8 **2**


2, -2 **2**


8, 4, -10, -2 **4**


2, 10, -10, -2 **4**


Here is what I've discovered:


-Running total will never go under 1, ever.


-Trends can lead into a long bout of 2s resulting into 20+ steps before it zeros out.


-Every running total starts with a positive even number


-So far, no number that is 6 greater than the current largest number is ever introduced. For example if 4 is the largest number, 10 can be introduced, but not 12. Not sure about this 100%, I'll need a more powerful device to test this theory.


-We can attempt to predict the numbers that are added or subtracted. Maybe AI can recognize a pattern if trained on the zeroing out data.


So using the above example


2,3,5,7,11,13,17


Can we use these rules to find the next prime? Well, yes, possibly:


The gaps of gaps is:
1,0,2,-2,2


So we have a sum of 3, but we want to get to a 1, this early in the sequence there is a high probability that the next gap of gaps is a -2. To achieve that you need to look at the gaps:


1,2,2,4,2,4


This would then require the next number to be a 2


Resulting in: 


1,2,2,4,2,4,2


Which the would mean the next prime could be: 19. Then we could test if it is, and yes it is.


Now of course it is entirely possible that the -2 wasn't the correct answer, but we have a range of possible numbers:


All even numbers from: -2 to 8


The reason for 8 has to do with 2 being the current largest introduced number and adding 6 to it. I'm not sure about this rule however. But I know it doesn't seem to explode and suddenly jump thousands, it's a bit predictable.


Any thoughts of why this property exists? Is it simply a bounded descent path?



Second Differences of Primes Pattern
For any sequence $p\_i$ with $\Delta^2 p\_i$ the second differences, then $$\sum\_{i=1}^n \Delta^2 p\_i = p\_1-p\_2-p\_{n+1}+p\_{n+2}.$$
So for $p\_i$ being the $i$th prime, this becomes $p\_{n+2}-p\_{n+1}-1.$
So whenever you have twin primes with $p\_{n+2}=p\_{n+1}+2$, the sum will be 1.


