Self-Similarity in Goldbach Sequences

We 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 self-similarity 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.

1: Complete Pattern Recursion: two Goldbach sequences have the exact same critical prime factors. The two patterns "mask" each other.

Ex: in the following two sequences, the prime factors 2,3,5,7 appear in the same step.
factors
hole zero 3 2 3,5 2,3
 
1 2 3 4 5 6
1 0 -1 -2 -3 -4
 
41 42 43 44 45 46
41 40 39 38 37 36

2 & 3: Incomplete Pattern Recursion (IPR) vs. Sufficient Pattern Recursion (SPR): We can make sequences using different step sizes. If one sequence has a step size of 1, and another sequence has a step size of a larger number, every critical prime factor will appears in the step in both sequences except the critical primes which are factors of the step size. Consequently, every place a hole appears in the sequence with step size 1, a hole will also appear in the expanded sequence.

Ex: in the following sequences, the prime factors 3,5,7 appear in the same step. 2 does not because the first sequence is expanded using a step size of D = 2.
factors
hole zero 3 (2) 3,5 3 (2)
 
103 105 107 109 111 113
103 101 99 97 95 93
 
1 2 3 4 5 6
1 0 -1 -2 -3 -4

The formulas for IPR and SPR are simply inverses.

IPR
Sn =P(p)*a/Dp +S1*Dn
SPR
S1 = (P(p)*a/DP+ Sn)/Dn

 

 

Introductory pages at this site

Parts of this page:

Definitions:
Sn: symmetry point in a goldbach sequence with step size n
S1: symmetry point in a Goldbach sequence with step size 1
P(p): the product of all the critical primes
a: an arbitrary integer that makes the fraction an integer also
Dn: the step size for Sn
DP: the product of the critical factors of Dn (eg: if Dn = 36 factors are 2&3, so DP = 6)

 More on Pattern Recursion

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 problem-solving (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.

 

Self-Similarity for Incomplete Recursion

The formulas above make it easy to determine where self-similarity for incomplete recursion occurs by letting Sn = S1.

Patterns will be self-similar when S = P(p)*a / (DP)(1+ Dn)

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 self-similarity 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 self-similar sequences will double.

Can self-similarity be used in a proof for Goldbach's Conjecture? Perhaps self-similarity 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 self-similarity and the same values apply to all sequences and period-doubling 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:
The formula we found for Goldbach self-similarity can be written as: P*a = S*DP*(1+ Dn)
The Logistic formula used by Feigenbaum was: y' = m*y(1-y)
It is tempting to conjecture a correlation between P*a and y', S and m, D and y. Although this currently seems unlikely.
 
 

Self-Similarity for Complete Recursion

Above we got incomplete self-similarity when the step sizes were multiples of critical primes. Taking a little insight from the form for complete pattern recursion, we can get complete self-similarity for step sizes that are not multiples of any critical primes. The starting point to find complete self-similar pattern recursion will occur when partition of all the multiples of all the critical primes differ by then even number in question:
G = a*Pa - b*Pb constructing G from its zeros
D = (a*Pa + b*Pb)/2 step size for S1 self-similarity

Here, step size will depend on G. We have not yet found a straight forward way to determine the correct self-similar step sizes for any given G with critical primes P. But we have demonstrated that each Goldbach sequence will have complete self-similarity. For sufficiently large critical prime lists, all Goldbach sequences will have multiple self-similar expansions.

By using modular arithmetic, we can map large numbers in the expanded self-similar 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 self-similarity 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
D = 1
1 2 3 4 5
1 0 -1 -2 -3
critical factors
hole zero 3 2 3,5
D = 769
1 770 1539 2308 767
1 --768 -1537 -2306 -765
modular mapping the cells in yellow have been mapped back into the sequence range using modular arithmetic around the pattern length (PL = 2310)
In the two sequences above the same critical factors (2,3,5,7,11) all occur in the same step in the sequence. We can see that the second sequence breaks the pattern into regions and maps those regions into each other in the first sequence. Consider that there are other self-similar step sizes and we get multiple mappings of a Goldbach sequence onto itself. This multiple mapping back into the pattern will prevent holes from bunching up in just one region, such that all the holes are outside the 0 to 1/G range.

Conclusions:

Our study of Goldbach's Conjecture has taught us some interesting things about pattern recursion and self-similarity. 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?

  • We have demonstrated that any list of critical primes will create factor pattern of known length holes having no critical prime factors with any Goldbach sequence. (pattern length )
  • We have shown that the mean distance between holes will be significantly smaller than G for large values of G. (MDBH)
  • We have shown that for small values of G, the actual number of prime pairs remains close to the number predicted by the mean distance between holes.
  • What we have needed is a demonstration that the holes will be somewhat evenly distributed so as to guarantee the existence of prime pairs between 0 and S = 1/2 G.
  • Self-similarity shows that sequences divide into various sections that map elements into other sections. This requires that holes in one section will map into other sections. Since multiple self-similar mappings exist, elements in any section map to multiple points in other sections. This will demand a somewhat even distribution of holes. (above)

 

G = the Goldbach even number = 2*S
a, b = arbitrary integer constants
Pa, Pb = products of critical primes where P = Pa*Pb
D = step size for self similarity
zero: a pair of number that together contain all the critical primes as factors
S1eq = a symmetry point that will have the same pattern recursion

 

Notes: Although Goldbach's Conjecture only considers positive numbers, for this study we examine Goldbach sequences from the symmetry point, S, the well into the negative number region so as to view the entire pattern. The core concepts of pattern length and critical primes exist regardless of what starting point we pick for our study.

Many of our examples use S=1. This may seem odd as it has only one sequence pair in the positive region. But it is useful for demonstrations since 1 is not a multiple of any prime. It is always a hole. Furthermore, the second element in the sequence is zero, which is a multiple of all primes. Hence we use the term 'zero' to denote any element in a Goldbach sequence which has all critical primes as factors.

 

 

 

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