# Possible contemporary improvement to bounded gaps between primes?

In his summary of his book Bounded gaps between primes: the epic breakthroughs of the early 21st century, Kevin Broughan writes

Which brings me to my final remark: where to next in the bounded gaps saga? As hinted before, the structure of narrow admissible tuples related to the structure of multiple divisors of Maynard/Tao, and variations of the perturbation structure of Polymath8b, and of the polynomial basis used in the optimization step, could assist progress to the next target. Based on “jumping champions” results, this should be 210. But who knows! [emphasis added]

We have a few years of computer and also presumably analytic development since Polymath8, are we able to push the bound down to 210 with current technology? What would it take, if not just someone with enough motivation to sit down and do it/spend the computational cycles? Are we still a long way from getting to below 200?

These are of course arbitrary numbers, in the grand scheme of things, since true progress down towards the bound of 12 enabled by the Elliott–Halberstam conjecture requires absolutely new ideas that seem too far from current knowledge. But there should be satisfaction in making some progress, even if not fundamentally substantial, similar to the recent case-by-case progress by Booker and Sutherland on outstanding sum-of-three-cubes cases below 1000.

• I'm myself eager to see some progress on this matter, so thank you very much for asking this question! May 7, 2021 at 16:02

I think that there is indeed some possibility to lower the bound, and this is something I've looked at seriously a few times. I spent a semester (in 2019) with the Computational Number Theory Group here at BYU trying to do it, without success. Let me outline some of the difficulties we found.

Issue 1. The main technique used in establishing the current bound is optimizing a quotient of integrals. This integral expression was first developed by Maynard and Tao (independently). The optimized integral quotient, in turn, can be approximated very well by finding the largest eigenvalue of a product of certain matrices, $$M_1 M_2^{-1}$$, related to those integral expressions.

To get better approximations, one needs to increase the size of the matrices. One of the roadblocks in improving knowledge of numerics is that the sizes of these matrices can be too large to store reasonably. I've spent some time optimizing code in Mathematica, storing the matrices as sparse association lists. One still runs into storage issues very easily, even in small dimensions. I'm happy to share this code with anyone who is interested. (Feel free to email me. Disclaimer: There are no comments explaining the code.)

That said, the real bottleneck is the next issue.

Issue 2. Generating the sparse entries that give rise to the eigenvalue problem takes time. To get to the current bound of $$246$$ in the Polymath 8b project we needed to slightly generalize the integral quotient mentioned above to an "epsilon-enlarged" region. This adds some complexity to the computations used to form the matrices. This epsilon-enlarged region is still amenable to quick computation, but the bound of $$246$$ might be the best one can get here.

The polymath project did go beyond the epsilon-enlarged region, using "vanishing marginals", and this led (after a huge amount of effort) to the prime gap bound of $$6$$ (under a Generalized Elliot-Halberstam conjecture). Unfortunately, these vanishing marginal conditions do not seem to be amenable to quick computation in medium-sized dimensions (or even small dimensions).

In the polymath 8b paper, there are different versions of the epsilon-enlargement idea, and perhaps one of these can immediately give an improved upper bound, but I've not been able to make that work.

To sum up: I believe that there is some small amount of improvement possible to the current bound, and this could easily be done if someone were to improve the epsilon-enlarged region in a way that is amenable to quick computations.

• I wonder if something like Charity Engine could be used (as the Andrews did for some of their sum of cubes work)... Or is it a problem with chunking the work/parallelisation etc? May 7, 2021 at 18:18
• I just looked back at the code I created, and it was better than I remembered. It uses an extension of the standard epsilon-enlargement (that appears implicitly in the polymath 8b paper). Issue 1 was the biggest issue after all--the association lists just got too big to fit in RAM. I don't know if parallelizing the storage of the association lists is feasible. It also might be possible to simplify something in my code to remove some of the storage. (I seem to recall that certain basis elements in the matrix formation were not as important during polymath 8b.) May 7, 2021 at 19:26
• ok, this seems like a reasonable target if people are looking to attack this problem. I guess the combinatorial data of the admissable tuples is not really an issue at this point...? As a suggestion, do you think your code is suitable to share and link to from asone.ai/polymath/… ? May 8, 2021 at 13:13