Goldbach's Conjecture, Pattern Recurrence, and Fractal Symmetry

In our last study we talked about masking patterns in Goldbach pairs. In this study we focus on the recurrence of those patterns between numbers. We also notice that the patterns recur at different scales of focus. This suggests that prime distribution has fractal symmetry of scale.


Written November 2008

Last Modified January 2010 (language improvements and links)

Part 1: Pattern Recurrence

For any number, there exists a set of other numbers that have the same masking pattern for the same set of critical primes. This is pattern recurrence

D1 Recurrents

The obvious masking pattern recurrence occurs when different numbers share exactly the same sequence of factors for a given set of critical primes. These we will call the D1 Recurrents.

In the diagram to the left we show masking pattern recurrence for the even number G=194. The symmetry point in the pairs is 1/2G = 97. The critical primes are 2,3,5, and 7.

The masking pattern is shown in the yellow box. The masking of multiples of 3 is shown in blue, 5 shown in red, 7 shown in green, and combinations shown in purple.

By looking at the pairs next to the symmetry point we can see that symmetry point 97, 83, 43, and 13 all have the same masking pattern. Also, symmetry points 8,22,62, and 92 have the same masking for 3,5, and 7.

3*5*7 = 105. This is a symmetry point for pattern recurrence of 3,5, and 7. The odd patterns and the evens pair as 105 - even = odd: Since 105-97=8, 97 pairs with 8, 83 with 22, 43 with 62, and 92 with 13.

The grouping of D1 recurrents appears to be related to the pairings in the masking pattern. The formulas for the D1 recurrents is conjectured to the right.
n mod(210)
n(5*7+3*3*4) mod(210)

n[(105-3*5)+105-7*2)] mod(210)

n[105-3*7)+105-5*40] mod(210)

Recall, this is for numbers smaller than 11^2=121.

Related pages at this site:


Part 2: Symmetry of Scale: D2, D4, D8 Recurrents

We may change our scale of view by 2^n. Each zoom will produce new numbers with the same masking pattern. We may also change our scale by any other number, but this requires us to ignore the factors of that number.

In the examples to the side we looked at the critical prime mask that corresponds to every other number (odds), starting from the symmetry number (1/2G). You should notice that this is the same critical prime mask shown above. So we now have 8 numbers, all with the same mask. So if one sequence, has a prime pair in the masking region they all will. Since we counted by twos to get matched pairs to the mask we call this the D2 recurrents. We may also show D4, D8 etc, numbers that have the same masking pattern as our starting number. This tells us that they all obey Goldbach's conjecture together because they all have the same masking pattern.

 

Now, we look at a given number's D1, D2, D4, and D8 critical prime masks for 194. The D4 mask occurs in two parts shift 0 includes the symmetry point, shift 2 is two numbers from the symmetry point. Similarly D8 has 3 positions (only one is shown.)

The colored numbers and colored underlines show the spacing between each critical prime factor and itself on the other side of the pairing. This distance sequencing is shown, color coded to the right. The same sequences will occur for the even recurrents above, but is a little harder to show since the symmetry points are not included.

From this we can see that 194, and all it's D1 recurrents, share many masking patterns with many other numbers. I suspect that if we include all the Dn recurrent masks we will find that all non-zero masks (no critical prime opposite itself) will be shared.

Thus for one number to be an exception many others, probably too many others, would also have to be exceptions. Goldbach's Conjecture would then have to be true. Note: the recurrent masks also carry very strong implications about where primes have to occur on the number line. That is, by using recurrent masks we can show where primes must pair up to produce a prime pair in the recurrent mask.

 

Part 3: Proving or Disproving Goldbach with Pattern Recurrence and Symmetry of Scale

To Prove:

  1. Show that Pattern Recurrence requires a smaller number to have the same masking pattern. The larger number cannot be an exception to Goldbach without the smaller number also being an exception.

To Disprove

  1. Show that pattern recurrence requires a masking length larger than G for some even number G.