
Introduction: Reviewing Pattern Recursion

Here we see 4 terms of three Goldbach sequences starting
from the symmetry point. The prime factors smaller than 11 are listed
at the bottom. The pattern is the same for each sequence. This is
the concept of pattern recursion. Multiple sets of sequences will
have same pattern of critical primes. The challenge, then, is to determine
whether numbers which share the same prime factor pattern, have useful
relationships that can be used to prove, or disprove, Goldbach's Conjecture.

Definitions:
 Goldbach Sequence: a series of number pairs that all add to a given
even number, G
 Goldbach Number (G): an even generating a Goldbach Sequence
 Symmetry point (S): the point around which the critical prime pattern
is symmetrical: S = 1/2*G
 Critical Primes: All primes being
considered in the Goldbach Sequence. Typically this would be all primes
smaller than sqrt(G).
 Number of Primes (n): how many critical primes are being considered.
 Hole: a number pair in a Goldbach
sequence not containing any critical prime as a factor. When both numbers
are smaller than the largest critical prime squared, the hole with be
a prime pair.
 P(p): the product of all the critical primes (eg: for primes 7 and
smaller, P(p) = 2*3*5*7 = 210. P(p) is the pattern
length for all Goldbach sequences with those critical primes.
Part 1: Complete Pattern Recurrence For any set of critical primes, every Goldbach sequence has pattern recurrence
with other Goldbach sequences. Each Goldbach sequence will have
2^n recurrents smaller than P(p) (including itself) where n = the count
of critical primes which are not factors of G. Every Goldbach sequence will contain holes. Thus complete pattern recurrence
may provide a means to develop a solution. The distribution of recurrents
may provide a method to determine a relatively smooth distribution of
holes, consequently a relatively smooth distribution of primes, as suggested
by Reimann&';s Hypothesis.
Method for Identifying Goldbach sequences showing pattern recurrence
with a given Goldbach Sequence
 Identify all the critical primes.
 Show all possible ways to break the critical prime list into two groups.
Primes which are factors of G must be shown in both groups.
 Multiply together the primes in each group.
 Find two integers which are multiples of those two products which
add to G
 Switch the sign of one of the numbers in each pair and add.
 This will give you more numbers that have the same factor sequence
as the number you started with.
This method may be written as the formulas
G = 2*S = a*Pa + b* Pb

 Pa & pb = partitioning of critical primes
 a,b = arbitrary integers to make partition add to G

S1eq = (a*Pa  b*Pb) / 2

 S1eq = the symmetry point of equivalent Goldbach sequences

Example 1: For ease, we pick G = 1 and Critical primes = 2,3,5,and
7.
a1 Factors 
2,3,5,7,11

2,3,5,7

2,3,5,11

2,3,7,11

2,5,7,11

2,3,5

2,3,7

2,3,11

a2 factors 
2

2,11

2,7

2,5

2,3

2,7,11

2,5,11

2,5,7

* 
















a1 
0 
2308 
420 
1892 
660 
1652 
462 
1850 
770 
1542 
1080 
1232 
1428 
880 
1122 
1190 
a2 
2 
2310 
418 
1890 
658 
1650 
460 
1848 
768 
1540 
1078 
1230 
1430 
882 
1120 
1188 
* 
















switched 
* 
4618 
838 
3782 
1318 
3302 
922 
3698 
1538 
3082 
2158 
2462 
2858 
1762 
2242 
2378 
G's 
2 
4618 
838 
3782 
1318 
3302 
922 
3698 
1538 
3082 
2158 
2462 
548 
1762 
2242 
2378 
S1eq 
1 
2309 
419 
1891 
659 
1651 
461 
1849 
769 
1541 
1079 
1231 
274 
881 
1121 
1189 
 We arranged all the Primes from 2 to 11 in the 8 possible combinations
where both groups contain 2 (since G is always even)
 We found integer multiples of these factor combinations that add together
to make G=2. Due to pattern symmetry , there are two numbers for each
combination. One with the pattern approach zero, one with the pattern
approaching P(p).
 We Switched the sign and added. This gave the Goldbach number where
the pattern recurs.
 We divided by 2 to find the symmetry points for each sequence.
Observations:
 The higher the number, the more critical primes, the larger the number
of evens with recurrent factor patterns. [2^n]
 However, the larger the number, the wider the spread of the recurrent
numbers [P(p)]
 To Prove: If for all large numbers, we can show the existence of a
smaller number with pattern recursion, then Goldbach's Conjecture is
proved.

Parts to this discussion:
 Reviewing Pattern Recurrence & Introductory Definitions
 Complete Pattern Recurrence
 Sufficient Pattern Recurrence on Scale
 Incomplete Pattern Recurrence on Scale
 Summation
Related pages at this site
Outside Links and References


Part 2: Scaled Pattern Recurrence: Sufficient Partial Pattern
(SPR)

Every Goldbach sequence may be viewed at multiple scales, or zooms,
Dn. Each scaled sequence will have pattern recursion with other
Goldbach sequences at normal zoom, D1. This is analogous to symmetry
of scale found in fractal geometry. Each scaled sequence will receive
sufficient pattern information from the D1 sequences it shares recursion
with. This will determine where its holes must occur. If a hole
occurs at a specific point in the D1 recurrent, then a hole must
occur in the same location in the Dn scaled sequence.
To the left, at the top, we see G=2 scaled by step size D = 3,
with its critical factor pattern below. Below that, we see three
recurrents with step size D = 1 which insert 3 back into the factor
pattern. Factor pattern holes are marked in pink. 
Each place a hole appears in one of the D1 recurrents,
a hole also appears in the D=3 scaling of G = 2 at the top. The S1
sequences sufficiently masks the S3 sequence. That means, where ever
a hole appears in any of the S1 sequences, a hole must appear in the
S3 sequence also. 
Definitions
 Dn: the size of the increment in a scaled Goldbach sequence.
 Sn: the symmetry point of the scaled Goldbach sequence
 D1 & S1: represent a Goldbach sequence with an increment of 1.
(no scaling)
Method for finding Sufficient Recurrents Formula: S1 = (P(p)*a/Dp + Sn)/Dn
 Dn: scale of zoom  a counting number
 Sn: Symmetry point of scaled Goldbach sequence
 P(p): product of all critical primes
 DP: multiple of all critical factors of d (eg: if d=24, critical factors
are 2&3, D=2*3=6)
 a: an arbitrary integer to ensure an inter result to the division
 any combination of a, D, and d with Sn and P(p) that produces and
integer S1 in this formula generates a new symmetry point, S1, which
has scaled pattern recursion which sufficiently masks the Sn sequence.
Observations:
 To Prove Goldbach's Conjecture: show that for any Sn, there is a smaller
S1, sharing pattern recursion with Sn, for reasonable choices of Dn.
 Reasonable choices of Dn appear to be on the scale of all counting
numbers smaller than sqrt(Sn).
 For every d, there should be 2D recurrents in range that fit the formula
above

Scale: Dn 
Prime Factors: D 
Recurrents 
2 
2 
2*2 = 4 
3 
3 
2*3 = 6 
4 
2 
2*2 = 4 
5 
5 
2*5 = 10 
8 
2 
2*2 = 4 
9 
3 
2*3 = 6 
12 
2,3 
2*6 = 12 
 For each prime factor product, DP, two S1 recurrents will exist in
each P(p) / Dn portion of the range.
 Every S1 recurrent will have 2^n D1 recurrents of its own (see part
1). The list may include redundancies.
 We still need to determine how many nonredundant recurrents there
are,or how many will be small numbers to use this method to solve Goldbach.

See demo in PDF 

Part 3: Scaled Pattern: Incomplete Pattern Recurrence (IPR)
Easier to work with, but not as solid for proofs, is the recognition
that each Goldbach sequence has incomplete recurrence with a set of scaled
Goldbach sequences. Places where holes occur in the scaled sequences,
holes have high odds of appearing in the original sequence. The scaling
constant will mask some of the holes in the S1 sequences. Thus, the hole
pattern is incompletely received from the scaled sequences.
Method for finding Incomplete Scaled Recurrents
This formula will determine a set of numbers providing incomplete pattern
recurrence for a starting number, S1.
Sn = P(p)* a/DP+ S1*Dn
Again, the arbitrary integer a is picked to ensure that Sn will be an
integer.



Summation and Observations
The observations above make Goldbach's Conjecture probably true. They
also suggest that the distribution of primes is rather smooth as predicted
by Reimann's Hypothesis.
 Every Goldbach sequence will have 2^n complete recurrents (including itself)
 Every Goldbach sequence will have at least n sufficient scaled recurrents
 each sufficient scaled recurrent will have 2^n complete recurrents
 Every Goldbach sequence will have at least n incomplete recurrents
 each incomplete recurrent will have 2^n complete recurrents
 Every sufficient recurrent may be associated with more incomplete recurrents
This expanding pattern recurrence with symmetry of scale would seem to
demand an smooth distribution of primes, as well as require that each
large number has sufficient recurrence with a smaller number. If this
is correct then Goldbach's Conjecture must be true, and Reimann's hypothesis
is probably true. However, to determine whether pattern recurrence may
solve Goldbach's conjecture will require some more work. 
