Goldbach Fractal Geometry

Introduction

Here we will look at examples of each form of recurrence, and discuss its implications.

Recursion Forms

Draft Jan 10 2014

1: Complete Self-Recursion

Every Number has complete pattern recursion with itself at multiple scales of magnification. This maps distant parts of the sequence back to near the symmetry point. This mapping requires holes to be roughly evenly spaced. It creates fractal symmetry of scale which tends to require holes to exist within the positive region of the sequence. Every sequence has multiple self-mapping with small step sizes, and an infinite number of self-mappings with step sizes larger than the pattern length.

 Example: To the left we see the beginning of the sequence of numbers that add up to 142 starting from the symmetry point of 71+71. Next to it we see the same sequence but with a step size of of 29. We can see that the pattern of critical primes is exactly the same. Two "zeros," points where all the critical primes are present are shown in yellow. Three holes, points with no critical primes are shown in gray. Modular arithmetic shows how -67+209 maps back to 59+83. Similar self-mappings for this number will occur with step sizes: 1, 29, 181, 41, 169, as well as all those numbers plus 210n.

2: Partial Internal Self Recursion

Every sequence can be broken into subsections that partially map onto each other. That is, all critical primes related to the subsection length will map out. This also tends to require the spacing of holes to be approximately even.

 Example: To the left we see the first 30 steps of the sequence of numbers that add up to 142 starting from the symmetry point of 71+71. Next to it,we see the next 30 steps. The sequence is almost identical except for the locations of the multiples of 7. Still, two holes occur in the same location. If we had picked a length of 42, every critical prime, 2,3, & 7 would have mapped the same, but 5 (not a factor of 42) would not have. Since the majority of potential holes in sequences get filled by by the smaller primes, considering subsequences of 30 or 210 provide strong structure for relatively small numbers.

3: Complete Recursion with Other Numbers
Every sequence also shares complete recursion with other numbers, for the same critical primes. As a result, holes will recur between sequences.

 Example: To the left we see the first 30 steps of the sequence of numbers that add up to 142 starting from the symmetry point of 71+71. Next to it, we see the exact same sequence of critical prime factors in the sequence of numbers that add up to 2. We can also find this same sequence of factors for 58, 82, 418, 362, and 278, as well those numbers plus 210n.

4: Complete Scaled Recursion with Other Numbers
Every sequence can be scaled by changing the step size to a prime number larger than the largest critical prime. This results in a new sequence that will map one sequence to another. This creates a situation where its highly unlikely that a first exception to the rule exists, because a number before it likely has the same pattern.

 Example: To the left we see the first steps of the sequence of numbers that add up to 142 starting from the symmetry point of 71+71 scaled with a step size of 11. The exact same critical prime pattern occurs in the sequence for 38. Conversely in the next example, 142 is shown scaled with a step size of one. This matches 52 scaled with a step size of 11. What we see is that sequence patterns recur with scaling between numbers which don't appear to have pattern recurrence. Here, we predict all sequences have scaled pattern recurrence with all other sequences where the symmetry point has the same critical prime factors. Scaled pattern recurrence with modular arithmetic mapping makes it highly unlikely that there can be a first exception to the Goldbach rule.

5: Partial Scaled Recursion with Other Numbers
Every sequence can also be scaled using a prime in the critical prime list which is not a factor of the symmetry point. When the symmetry point is a multiple of the critical prime, scaling may be done with an offset from the critical prime. This results in more mappings with partial pattern recurrence. As with partial self-recurrence, some of the holes in the sequence will be preserved.

 Example: To the left we see the first steps of the sequence of numbers that add up to 142 starting from the symmetry point of 71+71 with a step size of 3. Other than multiples of three the critical prime pattern matches 46 and 66 (major differences shown in pink and green.) Again, the hole pattern is mostly the same. Even shifting the second most common prime still leaves some of the holes the same in the partial pattern recurrence.

6: Rotational Symmetry Pattern Recurrence
Every sequence is symmetrical across its pattern length and the negative region of every pattern has rotational symmetry into the positive region of its neighboring patterns. As a result holes from one sequence will map onto neighboring sequences. With holes mapping rotationally between sequences it once again becomes unlikely that a sequence can exist without a hole in the positive region of the sequence.

 Example: To the left we see part of the sequence of numbers that add up to 210 near 0+210. The curves show how the pattern is symmetrical and the arrows show how the pattern rotates to map into other sequences. Since every sequence maps into sequences of larger numbers and every sequence is mapped into from smaller sequences holes in the positive region of the sequence become much more likely. (*Note: this example may be slightly misleading as the symmetry point occurs at zero. This does not happen for most numbers. The example was picked so that both rotational symmetry and sequence symmetry could easily be shown in the same picture.)

Related pages at this site

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Summation:

All these forms of symmetry strongly imply that holes must fill the positive region for all large sequences generated from large even numbers. This would prove Goldbach's Conjecture if the missing fractal details could be added. Until then, we can find beauty in the symmetries, including fractal symmetry found within Goldbach's Conjecture.