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You won't ever have a pair of 4's in a row. Let's say X is a prime number. We know it's odd as they all are except for 2. Every odd number is going to be either:
A) divisible by 3 (x mod 3 = 0);
B) have a remainder of 1 when divided by 3 (x mod 3 = 1);
C) have a remainder of 2 when divided by 3 (x mod 3 = 2).Let's consider X, X+4 and X + 8 when X is prime:
A) X is divisible by 3 - can't happen when X is prime
B) X mod 3 = 1
Then (X+4) mod 3 = 2
and (X+8) mod 3 = 0.
Therefore X+8 is divisible by 3 and is not prime.
C) X mod 3 = 2
Then (X+4) mod 3 = 0
Therefore X+4 is divisible by 3 and is not prime.So for any prime number X, either X+4 or X+8 will be divisible by 3.
Is that the reason for all the holes in the graph accept the stripes due to even numbers not being prime?
Prime nubers
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1st differenceCanceled comment.
Correction, the first difference is not 1 except between the 2 and 3 primes.
prime, 1st. diff., 2nd. diff., 3rd. diff.
2,1,1,
3,2,0,2
5,2,2,-4
7,4,-2,4
11,2,2,-4
13,4,-2,4
17,2,2,0
19,4,2,-6
23,6,-4,8
29,2,4,-6
.....
So the point for 2 is out of place, because the first difference is 1, unlike for the other primes. I cant figure out how to upload the image.
I got a pattern also using the second difference!!!! I searched to find your pattern.
It looks similar but is more regular in a square way the filled in parts. It makes a wedge and I think there are a few outliers from the wedge shape, but I think they are in the same patterned places as the pattern in the wedge.
Here is what I did. I took the first difference and the second difference and plotted them on each axis.
p_i is the i th prime. P_i+1 is the next prime.
fd_i=p_i+1 - p_i , first difference ,leading
sd_i=fd_i+1 - fd_i , second difference, leading
Plot points, x is fd_i and y is sd_i. This is the forward difference but works for the backward difference too. I played with the third difference in a 3D plot but this was the most interesting patern. There are bands missing and spots missing in a regular pattern.
You will never have a first difference of one.
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