Some problems with a theoretical attack strategy

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I'll use this issue page to list some problems which either myself or other erdosproblems.com participants have been able to identify as potentially being amenable to a theoretical analysis by someone with a suitable background in some existing fields of mathematics. Most likely these problems will require PhD level training in those fields to solve, though, and so this may not be a suitable project for crowdsourcing; nevertheless, I list them here for ease of reference, and to coordinate efforts and avoid duplication (in particular, I will update the status of problems from this list if I learn that they are being actively worked on already). Feel free to contact me by email if you are interested in working out the details.

• Problem 251. I noted that this problem may be conditionally solvable assuming a suitable form of the prime tuples conjecture, though there are a significant number of calculations to perform to actually achieve this. This recent paper of Pratt could serve as a model.

• Problem 387. I have an approach that splits the problem into two subtasks: firstly to construct a covering congruence system similar to that used in the study of large gaps between primes (Rankin type constructions), and secondly some sieve theoretic analysis to show that divisors in certain unwanted ranges do not occur. The latter can likely be sidestepped if one is willing to assume the prime tuples conjecture. I think this problem can be feasibly solved by someone who is familiar with sieve theory, probabilistic methods, and the theory of large gaps between primes. UPDATE: there is now a group of mathematicians working on this problem.

• Problem 421. I noted that many ranges of product lengths can already be treated by known tools or constructions such as Bombieri-Pila type bounds, the random deletion method, or anatomy of integers results. Some further work would be needed (with perhaps the injection of further novel ideas) to fully resolve the problem, but it does not seem hopeless.

• Problem 521 After some extensive discussion with Vjeko Kovac and Mehtaab Sawhney, it appears in principle that some combination of a switching strategy and the analysis of Yen Do could resolve this problem, though significant expertise in the theory of random polynomials would be required.

• Problem 689. This is again a covering congruence problem similar to that used in the large gaps between primes theory. After some conversations with Mehtaab Sawhney, we have arrived at a preliminary analysis that suggests that the linear equations in primes machinery of Ben Green, Tamar Ziegler, and myself, combined with some probabilistic constructions, may resolve this problem. This looks within reach of someone with expertise in the large gaps between primes theory, the linear equations in primes theory, and probabilistic combinatorics.

• Problem 848. This problem was recently solved asymptotically, but perhaps some combination of explicit analytic number theory and numerical methods can be used to convert this to a complete solution.

• Problem 884. I could solve this problem assuming the prime tuples conjecture. I could not find a way to remove this hypothesis, but perhaps one just needs to find a cleverer construction.