Thanks for both this comment and your reponse. It’s so interesting to glimpse insights into these (partial? Partial? PARTIAL? More than I could ever dream of!) results are realised, and how collaborations alight on them.

]]>The five of us are working on a related project involving exponential sums over primes that we hope to finalise soon. At one point in that project, one of us realised a possible connection of our work to Singmaster’s conjecture, although it turned out in the end that we were able to get our results using just the standard Vinogradov estimates on exponential sums on primes rather than the ones we have been working on.

The connection between equidistribution properties over primes (or equivalently, exponential sums over primes) and Singmaster started with the basic observation that when , the binomial coefficient is divisible by every prime between and , but not divisible by any prime larger than . This is already enough to get decent results in the regime (and was already exploited in an old paper of Abbott, Erdos, and Hansen), though for much smaller values of this is less useful as the interval is too short to provably contain primes. Our starting point was then the heuristic variant of the above observation (say in the region to avoid dealing with contributions of powers of primes) that tended to be divisible by most primes in but very few primes in , and this could be used to separate from , for instance if . This idea then went through several iterations and refinements (for instance we mostly started with first moment estimates on fractional parts and only later realised the utility of second moment estimates, i.e., correlation estimates) until reaching the current form.

]]>Yes, these two sampling methods are basically equivalent and which one to use is mostly a matter of personal preference. The weighting is often more convenient in proving the estimates, but the naive weighting involving primes drawn at random is conceptually simpler and is often the formulation used in describing many theorems or conjectures about the primes.

]]>I know you said in another comment that these arguments evolved from trying to generalize arguments that use the prime factorization of , but did you find Kane’s ”archimedean” approach first and then figured out a way to improve the proof with non-archimedean arguments?

]]>Also, to simplify the proof: you could sample your choice of for the application of valuation-comparison with probability . On the “back-end” (the actual computation of the correlations), this gets rid of the now-unnecessary summation-by-parts on (in the latest version as of the time of writing) the top of page 22, which serves as the transition from naïve uniform sampling to my proposed version of sampling; on the “front-end” (the application of the correlations to restricting solutions of ), this is perfectly okay because the failure probability of the identity is at most and the difference of the two sides in cases of failure is at most anyways, which introduces a perfectly-manageable error term of for the correlation equation.

]]>Nice observation! We were not directly inspired by the Bombieri–Pila determinant method (the argument we use comes instead from a paper of Kane) but now that you mention it, there is definitely a similarity that we will mention in the next revision of the ms.

This does raise the possibility though that it might be possible to use the determinant method (in either the Archimedean or non-Archimedean form) to improve the upper bound for the total number of solutions to ; the best bound currently is , due to Kane, and relying primarily on derivative estimates for this real-analytic curve. There the problem comes more from the edge of Pascal’s triangle than the interior, and I don’t think the rest of the methods in our paper are directly useful for this problem, but perhaps it is something for an expert in the determinant method to take a look at. (The key difficulty would be to get good quantitative estimates on the number of intersection points between the solution set and an algebraic curve of a given degree.)

]]>This is briefly discussed in the footnote on page 2. The main thing that is needed in our arguments are estimates of the form

for various large (this is roughly the same as asking that the fractional parts for prime are uniformly distributed). [Actually for technical reasons (involving the need to also control prime powers ) we work with a slightly shorter interval than and need a slightly more general phase and slightly stronger error terms, but never mind these complications for now.] The estimates of Vinogradov basically let us get something like this as long as , but pseudorandomness heuristics predict a much larger range of such as for some (perhaps even arbitrarily close to 1), which would lead to widening the applicability of our results to something like . In comparison, the Riemann hypothesis would (morally, at least) give results of the form

for such a wide range of , which certainly looks similar, but we could not obtain a direct connection between the two types of estimates.

]]>That sounds good, I’ll have to try to work it out!

Have you looked at whether one can improve these results (and possibly prove Singmaster’s conjecture) conditionally on other various conjectures (e.g. GRH to get better exponential sum estimates, bounds on gaps between primes to modify the regions where you work, possibly abc conjecture, etc.)?

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