SelfSimilarity in Goldbach SequencesWe have already looked at approaching Goldbach's Conjecture using Pattern Recursion. Now we ask, might Goldbach sequences be similar to themselves  i.e.: the patterns recur at different scales within the same sequence? Might selfsimilarity be an approach for solving Goldbach's Conjecture? Might we discover something else through this study? The first and third answers are Yes! But we still haven't found a solution to Goldbach's Conjecture. 
First Draft Jan 10, 2011 Last Updated April 10, 2011 

Background We have already identified three types of Pattern Recursion.

Introductory pages at this site
Parts of this page:
note to math teachers: various observers have suggested that one thing that makes math stale is that the false leads, errors, and uncertain conjectures that precede great developments in math, or even expert problemsolving (eg: engineering) get dropped from the discussion. Students only see mathematics in its final formal form. They don't get to see insights, false leads, and errors as part of the natural process. The links above ending with this page follow my independent studies through prime number theory. They don't show all my false leads and insights. They do follow the core sequence of noticing that patterns recur thorough attempting to formally describe those patterns so that the general form of patterns may be analyzed. 

SelfSimilarity for Incomplete Recursion The formulas above make it easy to determine where selfsimilarity for incomplete recursion occurs by letting Sn = S1.
This formula has meaning only when the denominator is a factor of the numerator. We quickly see that the formula can produce any value for S for given critical prime list P(p) and correctly determined step size Dn, by picking an appropriate value for a. Thus, all numbers will have the same step sizes for incomplete selfsimilarity for any given set of critical primes, P(p). From this we can demonstrate, for example, that all numbers will have self similarity for step sizes: 104, 105, 70, 69, 84, 85, 90, 91, 35, 36, 20, 21, 14, 15 where factors of the step size are ignored. (See attached.) Each time a new prime gets added to the list of critical primes the number of selfsimilar sequences will double. Can selfsimilarity be used in a proof for Goldbach's Conjecture? Perhaps selfsimilarity can be used to show that holes (potential prime pairs) must be distributed equally enough to require at least one hole to be in the region between 0 and S. Implications: Since we are looking at selfsimilarity and the same values apply to all sequences and perioddoubling occurs with each new prime, we expect Goldbach sequences to be examples of fractal geometry where Feigenbaum's Constants apply. We haven't determined how just yet. Coincidental Similarity:


SelfSimilarity for Complete RecursionAbove we got incomplete selfsimilarity when the step sizes were multiples of critical primes. Taking a little insight from the form for complete pattern recursion, we can get complete selfsimilarity for step sizes that are not multiples of any critical primes. The starting point to find complete selfsimilar pattern recursion will occur when partition of all the multiples of all the critical primes differ by then even number in question:
Here, step size will depend on G. We have not yet found a straight forward way to determine the correct selfsimilar step sizes for any given G with critical primes P. But we have demonstrated that each Goldbach sequence will have complete selfsimilarity. For sufficiently large critical prime lists, all Goldbach sequences will have multiple selfsimilar expansions. By using modular arithmetic, we can map large numbers in the expanded selfsimilar version back into the normal pattern length. This mapping results in a mixing of various regions of the entire pattern together. This mapping would seem to require that the distribution of holes is relatively smooth, and consequently prime pairs must occur within 0 to 1/2 G region. In short selfsimilarity would demand that Goldbach's conjecture be true, as both primes and holes are relatively evenly distributed. ex: consider critical primes 2,3,5,7, and 11, let S = 1
Conclusions: Our study of Goldbach's Conjecture has taught us some interesting things about pattern recursion and selfsimilarity. Our study has given us some tantalizing hints about the relationship between number theory and chaos theory, as well as hints how one might approach Goldbach's Conjecture. As Paulos has written, what we learn along the way will be more important than the eventual proof. Do we now have all the parts of the proof?



