Prerequisite:
As already indicated, our ultimate goal is to use dashed line theory to prove Goldbach’s conjecture. Dashed line theory itself provides more than one direction in order to arrive at a solution to the enigma; the proof strategies we are carrying out are illustrated on this page and in the related ones. However, these investigations are still ongoing, so their content is constantly evolving.
In this page and in the related ones, the statements not yet verified will be indicated with the wording Hypothesis and will be numbered using the initial “H”, so that they can be clearly distinguished from already proven properties; while definitions, properties and so on, presented in these sections, will be indicated with the letter “L” (from the Italian “Lavori in corso”, which means “Work in progress”). In addition, this nomenclature will be used:
Goldbach’s equation and pairs
- The equation p + q = 2n, corresponding to the problem underlying Goldbach’s conjecture, where 2n is the given even number and p and q are the two prime numbers to find, it will be called Goldbach’s equation.
- The pairs (p, q), which are solutions of the Goldbach equation, will be called Goldbach pairs.
The starting point
The main difficulty in proving Goldbach’s Conjecture is that prime numbers are involved, and this leads to a number of problems, that is why all the proofs attempted so far have failed. Our intent, however, is to try to get around the obstacle by involving prime numbers only indirectly, using something more manageable in their place. The tool we are using in our proof strategies is a particular type of dashed line, i.e. a table like this:
- The columns are the numbers from 0 to the even number 2n for which we have to search for Goldbach pairs;
- The rows, called components, are the first k prime numbers, where k, called order, is taken at pleasure;
- In cells such that the row number is a divisor of the column number, a dash is put;
- Columns that do not contain dashes are called spaces.
The interesting aspect of spaces is that they have a link both with the prime numbers and with the sum, and that is why our proof attempts are based, directly or not, on them.
For example, suppose we have to find two prime numbers that add up to 10. We can try with a first dashed line with k = 1, the spaces of which are highlighted in yellow:
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
2 | – | – | – | – | – | – |
In this dashed line we have three pairs of spaces which add up to 10:
- 1 and 9;
- 3 and 7;
- 5 and 5 (it is allowed to add a space to itself, because Goldbach’s Conjecture does not impose that the two addends must be different).
You would think that this is the right dashed line, because the last two are Goldbach pairs. But there is also a pair of spaces, the first, which instead is not formed by two prime numbers; so this dashed line cannot be considered completely decisive, also because, in reality, it only allows us to conclude that an even number can be written as the sum of two odd numbers. We can then try with a more “restrictive” dashed line (that is, with fewer spaces for the same columns), for example with k = 3:
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
2 | – | – | – | – | – | – | |||||
3 | – | – | – | – | |||||||
5 | – | – | – |
This dashed line, unlike the previous one, does not contain pairs of spaces which add up to 10: the only pair of spaces, 1 and 7, does not sum up to 10, and its components are not both prime numbers, since only 7 is. So this dashed line is not good either, because it is too restrictive, so as not to give rise to Goldbach pairs.
So let’s try something in between, i.e. the dashed line with k = 2:
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
2 | – | – | – | – | – | – | |||||
3 | – | – | – | – |
Unlike the previous two dashed lines, in this one all the pairs of spaces with sum 10 are Goldbach pairs: column 5, in fact, is a space and is also a prime number, 5 plus 5 equals 10, and there are no other pairs of spaces with sum 10.
Since we are looking for a universal solution, however, we cannot proceed this way, i.e. taking an even number 2n and proceeding each time by trial and error until we find the right dashed line: otherwise, we would be forced to construct a different proof for each of the existing even numbers, which is impossible, since they are infinite; our goal, however, is to find a single solution that applies to every even number.
Aiming to find some rule based on this example, we can already make a first series of observations on the dashed line which turned out to be correct for obtaining Goldbach pairs for the number 10:
- It has as components the first two elements of the sequence of prime numbers (2, 3, 5, 7, 11 and so on);
- If it had fewer than two components, we would get some pairs of spaces which have sum 10 but are not both prime numbers;
- If it had more than two components, we would not get pairs of spaces which have sum 10.
Therefore, the fact of having chosen two components seems to be an important constraint, so we could look for a criterion that tells us, given an even number 2n, what order k we should consider for a dashed line, having the first prime numbers as components, in order to obtain at least one Goldbach pair.
Choosing the right order
In order to hypothesize a rule for determining that k in our example must be 2, we can observe that, by Property T.1 (Spaces and prime numbers), the dashed line with components p_{1}, p_{2}, ... p_{k}, which from now on we will denote by T_k = (p_{1}, p_{2}, ... p_{k}), has spaces which are surely prime numbers if they are in the range from p_{k} + 1 to p_{k+1} ^ 2 - 1. Now, if we impose that the starting number 2n belongs to this interval, we have a good chance that the two spaces p and q also belong to it. In fact, for this to be the case, it is sufficient that 2n fall within the indicated interval and be as close as possible to its upper bound.
With this intuition-based reasoning we established a criterion for choosing k: it can be defined as the smallest integer such that p_{k+1} ^ 2 \gt 2n. In our example we started from 2n = 10, and indeed the smallest integer k such that p_{k+1} ^ 2 \gt 10 is precisely 2. In fact, for k = 2 the inequality holds, being p_{k+1} ^ 2 = p_{3}^2 = 5^2 = 25 \gt 10, while for k = 1 it would not be verified, since p_{2} ^ 2 = 3^2 = 9 \lt 10.
In summary, we started from the even number 2n of Goldbach’s equation and we found a criterion to calculate k, in such a way that the dashed line T_k = (p_{1}, p_{2}, ... p_{k}) allows to obtain the largest possible number of Goldbach pairs formed by spaces. In fact, by construction 2n is less than p_{k+1} ^ 2 - 1 and all the spaces of the dashed line T_k between p_{k} + 1 and p_{k+1} ^ 2 - 1 are prime, therefore they are good candidates for being substituted for p and q in Goldbach’s equation. Let’s formalize these concepts in the following definition:
Validity order, dashed line and interval
For each integer h \gt 0, the dashed line (p_1, p_2, \ldots p_h), having as components the first h prime numbers, will be indicated with the symbol T_h.
Given an even number 2n \gt 2, let k be the smallest integer such that p_{k+1}^2 \gt 2n. Then the integer k and the related dashed line T_k are called respectively validity order and validity dashed line relative to the integer 2n. Furthermore, the interval of the integers between p_{k} + 1 and p_{k+1} ^ 2 - 1 (both inclusive) is called validity interval of the dashed line T_k.
For example, let’s calculate what is the validity interval of our dashed line V = T_2 = (2, 3):
- p_{1} = 2, p_{2} = 3;
- k = 2;
- p_{k} = p_{2} = 3;
- p_{k} + 1 = p_{2} + 1 = 3 + 1 = 4;
- p_{k+1} is the prime number following p_{k} = p_{2} = 3, i.e. 5;
- p_{k+1} ^ 2 - 1 = 5 ^ 2 - 1 = 25 - 1 = 24;
- The validity interval is the set of integers between p_{k} + 1 = 4 and p_{k+1} ^ 2 - 1 = 24.
Property T.1 (Spaces and prime numbers), applied to the dashed line V, states that all spaces between 4 and 24 are also prime numbers, and vice versa. This is actually the case, because 5, 7, 11, 13, 17, 19 and 23 are the spaces included in this interval, abd they are also the prime numbers included in the same interval, so the Property is verified.
Search for Goldbach pairs
Methods based on the study of T_k
Now, however, the main problem remains: we know what is the correct order to find Goldbach pairs formed by spaces, but we can’t be sure we’ll find them, we have to prove it. We can translate this problem into a hypothesis:
Hypothesis of existence of Goldbach pairs based on dashed lines
Let 2n \gt 4 be an even number, and let k be the smallest integer such that (p_{k+1})^2 > 2n. Then the dashed line T_k contains at least two spaces p and q, such that (p, q) is a Goldbach pair for 2n.
We set 2n \gt 4, and not 2n \gt 2 as in Goldbach’s conjecture. Indeed, if we set 2n = 4, the corresponding k would be 1, being p_{1}^2 = 2^2 = 4 and p_{2}^2 = 3^2 = 9 \gt 4, so we should use the dashed line T_1 = (2); but in this dashed line the only pair of spaces with sum 4 is (1, 3), where 1 is not prime.
However, starting from 2n = 6, so far we have not found any even numbers that do not satisfy the hypothesis. Therefore, although it is precisely a hypothesis, i.e. it has not been universally demonstrated, it has not even been refuted. You can test it for any even number using the dashed line viewer, where there is a specific option for components, “Prime numbers up to the smallest p_{k} such that (p_{k+1})^2 > n“.
Based on this observation, it is possible to determine for example which are the even numbers associated with k = 1, \ldots, 5 and which are the corresponding T_k dashed lines:
Range of 2n | Order k | Dashed line T_k |
4 to 8 | 1 | (2) |
10 to 24 | 2 | (2, 3) |
26 to 46 | 3 | (2, 3, 5) |
48 to 118 | 4 | (2, 3, 5, 7) |
122 to 168 | 5 | (2, 3, 5, 7, 11) |
Going back to Hypothesis H.1 (Hypothesis of existence of Goldbach pairs based on the dashed lines), the next step is to try to prove it, i.e. try to understand under which conditions, among the spaces contained in the validity interval, there are at least two that have the starting number as their sum. In this regard, we are attempting to explore the following strategies:
Strategy using the properties of spaces
Strategy using the properties of dashes
Method based on the study of factorization dashed lines
There is also another way to use dashed lines to prove Goldbach’s conjecture, which considers a particular type of dashed lines, called factorization dashed lines. The difference with respect to the dashed lines seen previously is that the components of a factorization dashed line are the prime factors of a given number, in our case 2n. For example, if 2n = 10, the related factorization dashed line has components 2 and 5, which are the prime factors of 10:
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
2 | – | – | – | – | – | – | |||||
5 | – | – | – |
The advantage of using factorization dashed lines, compared to the T_k dashed lines seen before, is symmetry: as you can see, if p is a space, automatically q = 2n - p (Property L.F.2 ). For example, 1 is a space and 9 = 10 - 1 is also a space; similarly 3 is a space and so is 7 = 10 - 3. So, to find a Goldbach pair (p, q), we could start by finding a space p that is also prime, and then prove that q is also prime; but in proving that q is prime we could be somewhat advantaged, since it is a space by virtue of symmetry. We can therefore formulate the following hypothesis:
Hypothesis of existence of Goldbach pairs based on factorization dashed lines
Given an even number 2n \gt 6, its factorization dashed line contains at least two spaces p and q, such that (p , q) is a Goldbach pair for 2n.
Also in this Hypothesis we did not start from 2n = 4, as the statement of Goldbach’s conjecture does, but we had to skip 4 and 6 and start directly from 2n \gt 6, i.e. from 8. This is because the spaces of the factorization dashed lines of 2n = 4 and 2n = 6, respectively (2) and (2, 3), do not form Goldbach pairs; however the hypothesis, so far, always seems to be verified from 8 onwards.
Clearly, if Hypothesis H.2 were true, Goldbach’s conjecture would be a direct consequence of it. This gives rise to another proof strategy, the purpose of which is to prove Hypothesis H.2:
Strategy using factorization dashed lines
None of the mentioned strategies has yet proved to be a solution. These investigations are still ongoing, but thay have produced some partial results, which could lead to a final solution.