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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
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