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Section 1 - The Introduction:If you have been following along with this blog lately, you will know I started to document some of the mathematics JF covered in one of his episodes of The Public Space where he had developed an argument for the Goldbach conjecture. In addition to simply putting all of JF's work in writing and clarifying or otherwise elaborating some of the claims JF made on the show, I tried to make my own contributions. As I added more entries on the blog, JF used the posts to refine the argument even more, leading to the video you see above. I am not continuing the "JF's Theorem" posts in name, although I will still be adding posts on here about JF and my work on Goldbach under the new title "The Goldbach Airdrop". That way, we can differentiate between the older arguments and the newer ones. Of course, there will be references to the older posts throughout. Goldbach has always been a very elusive problem for a long time, and I concede totally that minds exponentially more capable than I have not had much luck in pinning down. Therefore, the goal of these posts is not to make grandiose claims that we have a foolproof way of demonstrating Goldbach right now, rather that we have an argument which could have a good amount of potential in generating an ever expanding pile of contradictions or restrictions if you assume Goldbach to be false. With that out of the way, we can begin with the foundation. Section 2 - The Foundations:In the previous approach, we had the D and T-sets: We will still be working with these forms of equations, as well as variations of the familiar D-set, under another name. So you can forget D and T having the same above meanings. E and R will still be defined in this way: The domain of discourse will be defined in this way: The R-set will be defined in this way: The R and D-sets are of course related. The R set can be called a proper subset of D: In place of the old conventions of the D and T-sets, there is a new setup for the S equations and how they relate to the R's: S_0 will be the same as the old T equation, and S_1 will be the D equation: A superscript denotes the R which is being referenced and the subscript greater than 0 references the power the R is being raised to: For this argument, we are trying to violate the Goldbach Conjecture. At about 14:20 in the above video, JF sets the constraints up such that Goldbach is not respected: To clarify why at least one of the operations in S_1, S_2, etc. need to be composite, consider the two operations and how they relate to each other: For the above example, suppose you had A = R^x. For S_1, it is the same idea. If the + and - operations both yielded primes, then you would be able to represent an E as the sum of two primes. You might have already considered that for S_0 and S_1, all R's are applicable, since they are only expressed to the power of 1, and therefore are still within the domain of discourse. When you move to the S_2 category, automatically some R's in the set, when squared, become too big and fall outside of the domain of discourse. A model to determine how many powers an R_x could be raised and still lie within D would look like this: We know that all of the R's can be represented in both S_0 and S_1, but we do not know how many will be represented on S_2 or S_3, for example. We know only that not all of them, and, as we raise to higher powers, still fewer R's to some power will still be in D. Section 3 - The Implications of Single Arrows:If you have already been following along with the progress made so far, you might recall from post 3 what JF called the "R-circles" which is a graphical way to represent the relationships between primes where the primes were vertices and the edges served as a way of pointing from one R to another. And from post 2, we proved that the graph cannot have loops. Working from about 21:33 in the video above, JF makes an important change to this model. Namely that an edge now is not simply a way of pointing between R's, but also the number of edges is now important. So we can now, for example, have R_2 pointing with a single arrow to R_5 in S_0 and that would now imply something much deeper: This will be a very important development moving forward, since it will allow us to determine which R's are still within D for an S greater than 1. Now, you might be wondering why a single arrow necessarily implies a power of 2 or greater. A reminder, we are trying to violate the Goldbach conjecture, therefore the arrow points to a single R, but if it were not raised to a power, you would be respecting Goldbach. Additionally, we had the important result from post 5 which proved plugging the same R into S_0 and S_1 implied that there were no common factors except in the + case of S_1 and then the common factor would be 3. Momentarily setting aside such a special case, we can say that the two graphs of S_0 and S_1 must be therefore distinct from each other. Now, if the two graphs show R's pointing to each other, and the two graphs are distinct from each other, then we can represent powers of some R's in two different ways: For the above example, in the S_0 case, R_1 leads to R_2, R_2 to R_3 and R_3 to R_4. This is a single arrow example, other quantities of arrows are possible as long as at least on arrow exists between two vertices, but the single arrow example is a critical casein point having very important implications. Now, for S_1, the vertices cannot point in the same manner as the first graph (again, discounting the special case) leading to R_1 pointing to R_3, R_3 to R_2, R_2 to R_1, and R_4 points to R_3. But what does that have to do with the bigger picture? At about 27:00, JF uses one of our major results from post 7, which was also further elaborated on in post 10. This leads to an important theorem: To obtain a form we are familiar with, the next step would be to factor out the common factor on the right side: Now how does that relate back to post 8? Well we know that if the contents of the parentheses on the right side is not equal to 1, we can draw the following conclusions: I think that is a good place to leave it for this post. The editor has slowed down a lot from the length and I am scared of losing all of this work. I will continue in the next post with where this conclusion leaves us in relation to the new approach to the problem.
Until later, - AF
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