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 6n-1 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

Minimun Holes in Pattern

Mean Distance Between Holes
Estimated Holes in Range
4
9
25
49
121
2
3
5
7
11
2
6
30
210
2310
1
1
3
15
135
2
6
10
14
17.1
2
1.5
2.5
3.5
7.1
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 = (2-1)(3-2)(5-2)(7-2) 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: (2-1)(3-2)(5-2)(7-2)...(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 (P-2) to (P-1). 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+d-2), 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

 

 

 
 
Other Number Theory Pages at this Site:
Pascal's Triangle
the golden mean
measuring compositeness
factor patterns
spatial number patterns
binary lessons
division by 0 lesson
base 12 chart lesson
 
 
More Number Theory
The Prime Pages
More on Goldbach
 
 

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 (2-1)(3-1)(5-2) = 6 instead of 3. Multiples of critical primes always have more holes than non-multiples.