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Last entry, I laid out some ground rules I said would become useful for JF's next claims. Specifically, for the things stated at 1:18:22 into his stream. The above video should start precisely then, so the reader can follow along well enough. JF's "R-circles":Again, for the sake of all of the following, we are assuming: JF says that R_2 is a prime factor generated from D_1, and it is put back into the search. This was something you might recall from the first post on JF's Goldbach work. The set of logical rules now comes into play. By rule 3, no R_x = R_x+1 and, at 1:18:49, JF says we cannot create a "self-conducting line". That is to say, an R which refers to itself. In graph theory, such a feature is referred to as a loop. Next, by rules 1 and 2, it can be deduced that all D's produced refer to a respective R. This corroborates JF's "R-circles". Amor Fati's Interjection:Now the R-circles is a really good development because it allows you to graphically understand that a D points to some R. We also know, as JF says, that "we cannot create a D that references it's own R". But, I believe I can give you a better representation with graph theory. For this, I will create what is called a bipartite graph. For a bipartite graph, all vertices of a graph, G, must be grouped into one of two sets, R and D, with no overlap. Further, no edges, E, in G can occur such that they exist between vertices of the same subset. Summed here: An important note is that all bipartite graphs are simple graphs. No simple graphs contain loops or multiple edges between vertices. This is where the rules in part II become useful. By rule 3, multiple edges cannot exist. We can deduce by the same rule that no loops may exist. Therefore, such a graph would be simple. By rule 2, we know that we can group vertices within two subsets, R and D, without any overlap. And again, by rules 3 and 4, we know that no edges will exist between vertices of the same subset. Hence, G can be represented as a bipartite graph: This will allow us to progress onto JF's Theorem II. So there is still more to come
- AF
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AuthorI'm just trying to learn about everything I can. Archives
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