OK, when you’ve finished the sieve argument I’ll try to go through it all thoroughly.

]]>Great news! I’ve incorporated this new info on the wiki and will shortly do so on the paper as well, and will also start a new thread focused on writing the paper, and clearing up any further loose ends. I think hitting the nice round number of k=50 (and hence ) is a nice place at which to “declare victory”; even though there may still be one or two more values of k we could squeeze out with enormous effort, it may be smarter to actually let things rest for a while in case some external development makes further progress a lot easier (similar to the situation with Polymath8a).

]]>Nice work! (it is still possible that the optimal epsilon is much larger – due to the fact that the optimal should vanish on a certain inner simplex whose size increases with epsilon – making it difficult to approximate by a single polynomial over the whole simplex.)

]]>I took epsilon=1/25 (I have no idea if this is optimal, but I believe it is close–others are welcome to run the code with different epsilons to test this). My basis of polynomials was of the form where is an arbitrary signature with only even entries of degree at most 26, and the total degree of the monomials is at most 27. The corresponding matrices have size 2526×2526, and the matrix formation step took almost two weeks. The generalized eigenvector step took 2 days; and we get . If people are interested, I printed two arbitrary entries from the matrices, so this can serve as a double-check if people want to verify that those entries are correct.

I next plan to run a similar computation using signatures of degree at most 28.

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On the newergap paper, I’ve uploaded to dropbox my source files for my write-up of the 3D computation. The main file is “BddGapsSize6.tex”, with the picture file “xyplot.pdf”. This version doesn’t contain some of the nice color graphics that an earlier write-up had, because I didn’t know at the time if we were going to simplify things further. (For instance, I believe that James showed we don’t have to break up the F-region any further.) However, I have uploaded the file “RegionPlots-BddGapsSize6.nb” which is the mathematica file that can be used to create 3D graphics. All you do is save the associated region plot as a pdf file, and then open up the pdf and re-save it in a condensed format (otherwise the graphic is too unwieldy). For brighter-better graphics, you can increase the plotpoint count. One can also remove the bounding box, the coordinates, etc… or add other information.

If this is too confusing, and there are specific graphics you would like me to make, I’m happy to do that.

]]>http://oeis.org/A091592 gives many counter examples

]]>1. there are infinitely many natural numbers. 1, 2, 3, ..

2. each interval of N^2 and (N+1)^2 has at least one set of twin primes

Is there a way to prove this?

]]>Either would be very helpful :). I’ve written down most of the sieving stuff that reduces the task of proving DHL[k,j] to that of performing a variational problem, but it was surprisingly lengthy, particularly the bit where one uses GEH to go outside the simplex. There’s one more section on that part still to write, because I’ve only computed the asymptotics when the multidimensional cutoff is a finite linear combination of tensor products of smooth functions, and there is a slightly tricky approximation argument to replace this with the case of piecewise polynomial cutoffs (which is what we use in practice). I’ll try to finish that off in a few days and then you could have a look at those sections. Then there are the sections in which we actually try to solve the variational problem, which have not been started on yet.. for the 3D stuff I remember Pace had some nice writeup here with pictures, maybe if he could donate his source files to the Dropbox then we could merge them in somehow.

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