
Goldbach's Conjecture
The Conjecture:
All even numbers larger than 4 are the sum of two primes. 
For example: 18 = 13
+ 5, or 102 = 97 + 5. 
This conjecture is simple
enough that a sixth grader can understand it or demonstrate examples,
yet the worlds best mathematicians have not solved it in over 200 years.
Math teachers:
all too often we fail to demonstrate to students the value of making mistakes,
and learning from false paths and divergent concepts. Sometimes what we
learn along the way is more important than what we intended to discover
at the beginning. Use this page to show what learning or new ideas might
occur from studying arcane conjectures such as Goldbach's.
The goal is either to prove Goldbach or to disprove Goldbach.
There is one obvious way to disprove Goldbach, simply, find one exception
to the rule. There is no obvious way to prove Goldbach, and other methods
of disproving Goldbach are not so obvious. I studied Goldbach's Conjecture,
but did not solve it. Neither has anyone else since Goldbach first proposed
it. But here are some ideas I stumbled on in the process of studying it.
Will any of these ideas help you solve it?
Some Important Notes about Primes
Critical Factors  The Largest Prime Needed
to Test a Larger Number for Primality
All composite numbers are multiples of numbers equal
to or smaller than their square root. Example: All composite numbers smaller
than 121 are multiples numbers smaller than 11, where 11 = sqrt(121).
Since 4,6,8,9, and 10 are composite, we need only test 2,3,5, and 7. Thus,
all numbers between 49 = 7^2 and 121 = 11^2 are either multiples 2,3,5,
or 7, or they are prime. So we may consider 2,3,5, and 7 the critical
prime factors for numbers in the range 49 to 121. This property is
well known in number theory. It will be needed for the ideas that follow.

If we list all the pairs that add to a given even
number, G, we have two regions to discuss. Numbers less than sqrt(G)
are in the critical region. In this region we find the critical
primes. Based on discussions below the region beyond sqrt(G) we will
call the masking region. 
The Density of Primes  Mean Distance Between
Primes
Knowing how many primes exist within a given region might
help solve Goldbach's Conjecture. Formulas have been created to estimate
this number. If we know how many primes exists within a range and we assume
primes in that range are spread out randomly we can estimate a mean distance
between primes. Mathematicians have created formulas to estimate the density
of primes. Here are some patterns
that may be used to help.
Location of Primes
All Primes except 2 and 3, sit next to a multiple of 6,
that is 6n1 or 6n+1. Thus, all prime pairs have a multiple of 6 right between
them for example: 5,6,7 or 11,12,13 or 29,30,31.
Methods to Understand and Possibly Solve Goldbach
Masking, Holes & Indifference to
Factor Arrangement
In any region, each composite is a multiple of at least one critical
prime. Multiples of each critical prime create a simple pattern.
Together they create a complex pattern. Thus, to study pairing patterns
we can look just at the patterns created by the critical primes. That
pattern creates masks that prevent primes from pairing with
primes. Spaces not masked by critical primes are holes in the
pattern where prime pairs occur.

Here we see the masking pattern
for 32. The maximum masking length for 32 is 11 since all numbers
from 2 to 11 are masked by 2, 3, or 5. However, since the pair
3&29 is in the critical region it is still a prime pair
even though it is masked. Green marks the holes and prime pairs,
purple marks the end of the critical region. 

To Prove: Show that there will always be a hole in the critical
factor mask somewhere between 0 and G (the even number.) 
To Disprove: Show that the critical factors will eventually
mask the entire region from 0 to G. 

In the diagram, we see part of a masking pattern
for 2, 3, and 5. The first space is a hole since it is not
masked by 2, 3, or 5. A prime pair will occur in that hole. The same
pattern will mask all numbers not multiples of 3 or 5. For numbers
larger than 49, larger primes starting with 7 must be added to the
masking pattern. Here, we show how this pattern masks 32, 38, and
42. 
The diagram shows how masking is indifferent to which side of the pattern
factors occur on. We can see how although the masking pattern is the same
for different numbers, the masking factors occur on different sides of
the pattern.
Pattern Length (1) Number Line  approximate density
of primes for a region between squares
Multiples of a set of numbers create a regular repeating
pattern. The length of the pattern is the product of all those numbers.
For example the multiples of 2,3,5, and 7 create a pattern that is 2*3*5*7=210
integers long and that pattern will repeat every 210 integers. (More
detail)

Here we see how the factors 2,3, and
5 create a pattern that is 2*3*5 = 30 integers long. The Factors are
red, blue outlines show multiples of 6, and green shows that primes
must occur right next to multiples of 6. 
30 is the last number where pattern length is smaller than the range:
P(n+1)^2. This will make it much harder to use pattern length to solve
Goldbach.
Masking Length
Masking length is the distance between adjacent holes. Maximum masking length
is the longest distance between holes that a set of critical primes can
achieve. If the maximum masking length is shorter than G (the even number
being tested) then a hole  a prime pair must exist that adds to G. Unfortunately,
finding a rule for maximum masking length is very difficult, possibly impossible.
So, I studied mean masking length instead. If the mean stays small with
respect to G, then the maximum masking length might stay less than G. Then
Goldbach would be true.

32 has a largest masking length of 12, from (1,31) to (13,19). This
is too short to mask the entire range of 32. 
To Prove: Prove that the maximum masking length
will always be less than G. 
To Prove: Prove that the ratio between maximum masking length
and mean masking length is always less than the ratio between range
and mean masking length. 
Mean Distance Between Holes and Estimated Holes
in Range
For any (small) range we can determine the critical factors, pattern
length that goes with these factors, and the minimum number of holes that
will occur in this pattern. From this we can determine the Mean Distance
Between Holes and use this to estimate the number of prime pairs within
our range.
Range Minimum

Largest Critical Prime

Pattern Length


Mean Distance Between Holes

Estimated Holes in Range








For numbers between 49 and 121, the
largest critical prime will be 7. The critical primes up to 7 have
a pattern length of 210 = 2*3*5*7. This pattern will have 15 = (21)(32)(52)(72)
holes. This gives a mean distance between holes of 14 = 210/15. For
the smallest even 50, that gives 3.5 = 50/14 holes in range. 
Implication: since the range rises much faster than the mean
distance between holes for the same critical primes the number of
prime pairs will tend to increase as magnitude increases. This makes
Goldbach's conjecture quite probable. 
The minimum range being examined will always be the square of the
largest critical prime: P(n)^2. The Pattern Length will always be
the Primorial of the largest critical prime: P(n)! Since 2 masks itself
and every other critical factor will mask 2 numbers in its own length,
the holes in a pattern will be in the form: (21)(32)(52)(72)...(P(n)2).

When ever the even, G, is a multiple of a critical factor, P, then
P's part in the multiple would be adjusted from (P2) to (P1). Thus
numbers not in the form 2^n will have more holes, and a shorter mean
distance between holes. 
The first graph shows that for small numbers the actual number of prime pairs remains close to the number of predicted holes. The second graph adds to that by showing that, for small numbers, the minimum number of holes gradually steps up as the Goldbach number increases. Can this demonstration be extended to all large numbers?
To Prove: Prove that the actual number of holes (prime pairs) in
the masking always remains larger than, or close to, the mean
number of holes. 
Implications:
 If Goldbach is false, the first exception is most likely a power
of 2: 2^m, or 4 times a large prime: 4P
 Exceptions to Goldbach most likely do not include multiples
of small primes. E.g.: 3 can mask 2/3 (66%) of all numbers when
G is not a multiple of 3, but only 1/3 (33%) when G is a multiple
of 3.


In the graph, we see the number of masking
region prime pairs for each even number (blue) up to 160. This is
compared to our mean estimate (pink). Both tend to rise, and neither
drop to 0. 
Boundary Value Conditions
In physics, boundary value conditions for a problem frequently lead to a
solution. Goldbach's conjecture has two boundaries that set limits
on how masking may occur. The first condition is the symmetry of the
masking pattern around 1/2 G. The second condition occurs from where
zero pairs with G through the critical region. 
Boundary Condition #1: Masking Symmetry of Goldbach Pairs

To Prove: Show that symmetry requires a hole to occur in
less than 1/2 G from the point of symmetry. 
To Disprove: Show that symmetry requires holes to spread
out faster than the growth of G. 
In a list of pairs that add up to an even number, critical factors
will be symmetric around 1/2 the even. Thus critical factors must
mask the entire length from 1 to G.
For example, when testing 26, 13+13
= 26, then 12+14 = 26, but so does 14+12. Regardless of the order
the numbers we add have the same critical factors, 2 and 3.


Boundary Condition #2: Zero and the critical region
[a] Zero always pairs with G. Zero is a multiple of all numbers.
So all the critical factors in the masking pattern must mask the pair
[0,G]. Similarly, all factors of G mask [0,G]. They are not offset
as other factors. 
[b] The masking pattern in the critical region must include the
normal sequence of primes. Unlike the masking region ( sqrt(G) through
1/2G ), in the critical region the critical primes do not mask themselves.
They must be masked by another element in the masking pattern. For
example, if our number is 256, the critical region is:.
3, 5, 7, 11, and 13 are in the masking pattern but they are prime
so they do not mask themselves. They must be masked by another number
between G, 256 and 240. 
To Prove: Show that at least one critical prime must be unmasked
for large values of G. Or show that whenever all the critical primes
are masked at least one hole will occur in the normal masking range. 
For a small set of primes it will be unlikely that the pattern can
mask critical primes without being a multiple of some of those primes.
Whether this holds for large sets of primes is not clear. Stated another
way: Subtract each critical prime from G. You will either get a multiple
of a critical prime, or you will get a prime. Each new critical prime
added to the list gives one more chance that a critical prime will
pair with a large prime. 
Implication: Since each new critical prime increases the
probability that a critical prime will remain unmasked, Goldbach is
probably true. 
Holes & Hole Filling

When 3 is the largest critical factor, the holes
in the pattern occur in regular intervals of every sixth integer
pair. This is the last time the hole pattern will be regular.
With each new prime complexity in the pattern increases. 
However, we might be able to build on the effects of this regularity.

To Prove: Show that the critical factors larger than 3 can
not fill all the holes in range from this regular spacing. 
The effects of variations in prime number spacing
We have seen above how holes, pattern length, range, and mean holes per range,
vary with each new prime added to the list. We can organize our thoughts
on these items and check what happens to mean distance between holes as
each new prime is added to the list.
Let's look at the nth prime, Pn. We can call the holes in the pattern
Hn, the pattern length Ln, and the range Rn. Then we can go to the next
prime Pn+1 = Pn+d. For this new prime we have holes: Hn+1 = Hn*(Pn+d2),
pattern length Ln+1 = Ln*(Pn+d), and range RN+1 = Pn^2. When we put these
together to get the mean distance between holes, we can rearrange our
terms to get: Mean Holes in Range = Hn/Ln[(Pn+d)^2  2(Pn+d)] . In this
equation the positive parts rise as a square of both the previous prime,
Pn, and the distance, d, to the new prime. The negative parts rise linearly
with both. So, for large primes the holes in the pattern will tend to
increase.
Implication: Since the mean holes in range will tend to rise with
each new prime added to the list, large even numbers will tend to have
more prime pairs than small. Thus Goldbach's conjecture is probably true.

First Draft February 2003
Current Draft: Minor changes March 2011





Note: If G is a multiple of a spefic prime, then we subtract 1
instead of 2 to determine the number of holes. eg: Since 24 is a
multiple of 3, the number of holes for 24 would be (21)(31)(52)
= 6 instead of 3. Multiples of critical primes always have more
holes than nonmultiples.




