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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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In post 7, we went through the proof by cases, which was a contradiction of R_x or R_y being greater than a common factor, c, in the stream at 1:39:50. JF does suppose that c will be a prime, and I had also reasoned that in post 7. But, after discussing it with him, we realized that it could just be a trivial assumption, and not necessarily true. So how can we know that c is actually prime? It could be because I have not slept in two days, or that the approach was not intuitive for me, but I, after quite some time, believe we can demonstrate the c is necessarily prime. I also wanted to elaborate on a conclusion I had reached before in post 7, which I will do first. This special case of z = 1 is applicable in the two cases: And, from the work we did in post 7, we know: The problem was one of the conclusions I reached, which was that if z = 1, then 2D_x > T_y. That is true, but I felt the need to mention that this does not necessarily imply that 2R_x > R_y. That is because there were two cases where z could be 1. That is all, but it could lead to some possible misconceptions. With that out of the way, what is a method we can use to determine that c is a prime? We can employ the proof by contradiction again. Suppose that c is comprised of prime factors such that c = ab. So 2D_x will be divisible by a, b and c. Perhaps you could make the case that if: But then you would need to recall: This leads to a problem already. First, neither a nor b would not be applicable, since z must be 1 for the case to work. But, then the factor, a, would be a common factor for 2D_x - T_y and also R_y +/- 2R_x. You can apply this reasoning and you will arrive at the contradiction: So I think we are safe to say it, JF. We have z-primacyI will add more content later. Keep your eyes out, lads.
- AF If the past few parts have given us anything important to discard form the original two equations, it is the plus case for the D-set. All weird exceptions have arisen from this algorithm, so let us, for a second, discard it and look at the implications of only having the two equations: Next, for a kind of common ground to derive some kind of relationship between the two equations, we can let R_x = R_y for the first iteration. I will adopt the convention of the x for the D-set and y for the T-set, with a number following showing how many iterations have taken place. To make it clearer to the reader, for the first iteration, I will make the two R's equal. This is demonstrated here: In fact, we can adopt a new set of equations which will get the same idea across, but be easier to work with: And as long as the original E is still divisible by 4, we should be getting the same results. Like I said, make both initial R_x and R_y equal: This next part will be a bit messy, but I will list a few examples I did and we can find any relationships between T and D:
A second example:
And a final example:
If you play with the relationship between the respective D and T iterations, you will notice that it appears a pattern emerges: This can be proven simply enough: If you take other iterations, it seems like the only things generated are prime numbers, related to their respective R_xn and R_yn numbers. (Look at example two, and apply the same idea with R_x2, R_y2, etc. if you want to see those interesting results.) And now that we can put all of the cases where things get weird in the plus case of the D-set, I can implement the rules found in post 2 and post 6 before. I will also try to set up something like JF's R-circles from post 3. We can actually determine a good amount of information from these two things, with the minus case in the D-set, and where R_x1 and R_y2 are equal. I apologize for the bad quality of drawing, but I have grouped the R_x's in blue, the R_y's in yellow, and I will use a green line to show a possible line of equality. For instance, we know that no R_x1 = R_x2, and no R_y1 = R_2, so they were colored red. The R_x1 = R_y1 is green since we have started with the two values being equal. And here we arrive at one of our first controversies: can an R_xn = R_yn where n is not 1? Some of my progress on this: It is a long way of saying "I do not have an answer yet". The truth is, I believe that, because there is no contradiction in assuming some other R_yn = R_xn, we can probably assume that some exceptions exist. So let us assume out last graph and R_x2 = R_y2, and look at the implications: It is also already implied, but this is a sort of case in point for when more vertices are added. For another example: If R_x3 = R_y3, then it is implied by what we already know from the posts before that R_2y != R_3x and R_2x != R_3y. I will lay this out logically for clarity:
But what about if an R_xn+1 = R_yn, or an R_yn+1 = R_xn? Well, take a look at the graph now: Now, R_x3 and R_y3 cannot be equal, since it would lead to a contradiction (R_y3 would then equal R_y2). Second, you could not have an R_x2 = R_y2, since you would lead to the same contradiction on the other side. Finally, big picture, there would never be a loop back to R_x1, since then you would need to say R_y1 = R_y2, which is also impossible. This leads to a very important theorem when looking at these relationships between R_x's and R_y's, which could be very important to the proof: You could swap x for y, it works vice versa. But if there is a way to show that a loop can never occur, and since the number or R_x's and R_y's must be finite, then there must be a point where you get two primes adding up to 2E.
There is still much work to be done, - AF Last entry, I drew out the whole proof by cases to prove that if a D_x and T_y shared a common prime factor, c, then either R_x or R_y would be greater than c. Of course, we demonstrated in the contradiction, that a special case, z = 1 existed. This entry, I will extrapolate on the "things we can conclude" section on last post. Recall the following equation: And, setting aside the special case of z = 1, which we showed last time will produce a prime number, what can be discovered for when z is not 1? So, only when z = 1 is R_y +/- 2 R_x is prime. Also, if this is not the case, then cz will not be prime. And since z is not 1, and can be expressed by a fraction, then 2D_x - T_y will not be prime. It will likewise be divisible by c. Also, R_x or R_y will be greater than c. And, of course, it is also implied that R_y +/- 2 R_x will not be prime either.
Some interesting consequences, maybe applicable to the development of the argument. Until later, - AF Pardon for not making a timely update to the progress through JF's segment on his work through trying to make headway into Goldbach. But I do not yet totally understand the rationale for JF's looping argument. So I have been kind of stuck for the past few days, making little headway. That being said, I figured I might as well give the progress I have made so far, and if anything more comes of it, I will amend it. This way, my work does not stagnate and die. I am going to be working on the content from about 1:39:50 in the stream. JF says he would like to prove the following: Having the general equations for the D and T-sets: And supposing that D_x and T_y share a common prime factor, c, then either R_x or R_y is greater than c. The problem, JF says, is it is not true. And he is right. Supposing they did share a common factor: We will set up a proof by contradiction, and will need to work by cases. For the cases such that R_x and R_y are not greater than c, we have:
And need to check these values for R_x and R_y for +/- 2 R_x
We also can conclude that z is odd. Therefore, you could have a z such that z = 1.
This violates JFT, of course, so we can conclude the opposite, that is there will never be a z such that z = 3.
Both cases yield the same type of contradiction. Makes sense, since this is a violation of JFT.
So we now know that z != 3, and z > -1. This establishes an upper and lower bound.
If R_x and R_y = c, then z = -1. This is likewise a violation of JFT. Maybe this was not the best way to present, because we got the same idea from the previous case. But I am being thorough. So we have examples where z can definitely exist and R_x or R_y are not greater than c. But we have also shown it is for one specific case. Now we can list some important conclusion here:
That is it for now, sorry for the delay. - AF I have been getting lost in the weeds lately in JF's Goldbach conjecture argument. To such an extent that I believe there could be some fundamental issues JF may have overlooked that doom it. Or, what is probably more likely, I am missing something. After all, JF's stream was pretty rushed, demonstrating a lengthy, but elegant set of arguments. Therefore, I want to perhaps explore the T-set in more detail and maybe connect the two constraints with what we now know from previous posts. From post 2 , I proved that, for the D-set, the following logical rules were true: This can be proved identically for the T-set. Again, the assumption we have been working with must be true for the rules to be practical. The assumption being, T is not prime. For the sake of clarity, this post will seek to demonstrate to the reader the following:
By JFT, we know necessarily that T and E are coprime, so there is a contradiction.
Let us see where this leaves the rest of the proof. I have some ideas to elaborate on from here and what we already know.
- AF Last entry, I gave the expanded proof by cases of JF's second theorem, which implied that if two D's, D_x and D_y, shared a common factor, c, then at least one of the other prime factors, R_x and R_y, would be greater. For this entry, I will be skipping a section which, incidentally, JF says is very important. I plan on getting back to it later after I understand all of what JF's work does. This entry will correlate with the contents of the stream starting at about 1:29:37, which should be when the above video begins. This is where JF introduces what he calls the "T-set". Recall the equations we have been examining previously. The equations of the form: And, again, we are supposing the set of D's produced consist of prime factors which can be put back into the equation above, the reiterative search for a pair of primes. Now, we can slightly modify this equation wherein we can likewise introduce these corresponding R's into. This is what produces the T-set. Here is that equation: Now, we know by JFT for the D equation that D and E will always be coprime. As you can probably guess, it is the exact same demonstration which shows that T will likewise always be coprime to E: Like the proof before, R must be prime, which implies no R/C exists which is an integer, and since B must be an integer, we can conclude the contrary is true. Therefore, T is always coprime to E. Now it has been established that both T and D are coprime with E. The next natural question you would have to ask is whether or not T and D are coprime with each other. In the stream, JF says that they are coprime except for if they have a common factor of 3, or in the "+" case. JF has a different proof, but I will demonstrate to the reader now some of these truly interesting results which were on the stream in more detail: This will lead you to two expressions for R: Expression 1: Expression 2: Now, here is where things get interesting. We can do the same type of demonstration on expression 2 as we have been doing many times before. That is, assume that T and D have some common factor C. By letting T= AC and D = BC: Another spicy contradiction. But what does this imply? In expression 2, since R is positive, we can conclude that T > 2D. But this may not always be the true. A case in point: Suppose E = 224 and R = 17. We will use the addition side of the D equation:
So, were you to input these numbers, all of which satisfy the constraints, into expression 2, you would end up with a negative number. This means that the two results should be treated as cases. Expression 1 is the case for when 2D > T and expression 2, conversely, when T > 2D. This leads us to make some very curious implications: So, let us suppose you have case 1, where 2D < T: you know T and D necessarily share no prime factors. For case 2, we can actually use my above case in point: 129 (mod 3) = 0, so T, which is 207, will likewise share the common factor of 3. (For the record: 207 = 3 * 69). For case 3, I can offer you another case in point:
And neither 149 nor 187 are divisible by 3. But 2D-T will always be divisible by 3. Next, I will show you a series of three graphs: We can draw some important facts looking at these three graphs. All are of the same forms as the respective either D equations or the single T equation. Facts we have found:
That is, the only time you will ever find a common factor between a D_x and a T_x is when D_x is found from: E/2 + R_x = D_x. And R will only be in a domain of (0, E/4). Not only will those two facts be true, but the only common factor they will ever have will be 3. Wow. That does not imply all primes in this domain yield D's and T's divisible by 3, but if they exist, they can only be found there. A similarly important fact which is not implied here is that a D_x cannot share any common factors with a T_y. I say again, that is not necessarily true, they could potentially share common factors in that case. Maybe D_2 shares a common factor with T_4. That is a post for another time, though.
I was proud of this post. Seems like some good information could be within it. Until later, - AF Last entry, we showed that JF's R-circles method could be represented in the form of a bipartite graph consisting of two subsets: D and R. The next theorem JF gives us can be derived, actually, using JFT. Because Weebly seems to be incapable of giving you the exact time I stamp in the YouTube URL, JF starts work on theorem 2 at 1:20:27. For a prime factor shared between D_x and D_y, C, either R_x or R_y is greater than C. I will put this also in the LaTeX: Supposing D_x and D_y have a shared prime factor of C, and taking the difference between the two equations (JF does D_y minus D_x, so I will follow this convention), we get: Since we know the result will be an integer, let everything in the parentheses on the left of the equation be z: Now, for the sake of thoroughness, I will do the proof by cases more in-depth than was presented on JF's stream. We will again set up a proof by contradiction by supposing the three following cases: Case 1: Same result as in post 1: JFT. We know already that z must be an integer, and R_x is a prime. Case 2: I added the equation just for the sake of redundancy. This results in the same quagmire as case 1. Case 3: This likewise leads to a contradiction. Therefore, at least one of the R's must be greater than C. Amor Fati's Interjection:You might have realized a possibility is R_x = R_y = C as a case. But if they were equal, then there would be no difference, and therefore no point to the theorem. In addition, we already know from JFT that there exist no cases ever such that D_x has a factor or R_x. These cases were added for the sake of redundancy. Also, this theorem supposes that there exists a common factor between two D's. For what it is worth, I am not totally convinced such a thing is possible. Granted, I have no proof. That is part of the reason I went into so much detail about the bipartite graphs, since there is a corollary to bipartite graphs which goes something like: No bipartite graphs consist of an odd cycle. That is, all bipartite graphs are even cycled. But this does not quite pan out as well as I had expected. Because you could assume that something like this bipartite graph occurs: Now, I still left the rigorous explanation for the aforementioned reader who is serious about utilizing JF's work. My initial line of reasoning was that an even number of D's leads to an odd number of R's, and if there is an odd number of R's and they cycle, then there would be an odd number of vertices and that would prove that no D can reference to a previous R. But that was really naive and a foolish instinct, because you end up with just the type of graph above: and even number of R's to D's, and, what's more, an even number of even R's to D's. So you cannot really make any progress there, at least I cannot. But, hey, I leave that stuff in for if there is some progress to be made in something I had already thought about. This is supposed to be a resource you can get JF's work from, and some potential avenues for progress on a yet unsolved problem.
That is all for now, -AF 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 Last entry, I made an organized display of JF's Theorem (henceforth on here "JFT"). I said he continued his proof, and I wanted to make a series of entries here so that the reader wasn't inundated with information. Next, I want to expand on what we have already learned before we get to theorem two in JF's livestream. The next thing JF says is you cannot infinitely iterate D_x's. This is true, since E is a finite number. He makes some claims next that I will not verify in this post. But I think I can use some simple logical rules to lay a groundwork for his next theorem he presents in the stream. This is the case for when: And here are the logical rules I can give you:
To prove no R_x = R_x+1, you would have to suppose D_x was coprime with E, but not prime. (JFT) Suppose also that D_x has factors, A and C. If an R_x+1 could be generated in such a case, then R_x+1 would have the same value as R_x. For a proof by contradiction, one of the prime factors would have to be R_x. To put it more generally: So let A = R_x, and we can solve for C: And how does such a result relate with what we already know? Well, by the constraints, we know that E and R_x are coprime. We also know that C is an integer. The only way the left side of the equation will yield an integer is when R_x = 1. You are probably saying the proof looks familiar, because it does. This is just JFT, applied with a prime factor of R_x.
Proof of 4: Consequence of 3. Since, if, besides 1, no D_x and D_x+1 share common prime factors, then D_x and D_x+1 cannot be equal. I think this can lay the groundwork for some of the claims JF makes next. -AF |
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