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A crowdsourced project to link up erdosproblems.com to the OEIS

31 August, 2025 in admin, advertising, Mathematics, polymath, question | Tags: , | by | 19 comments

Thomas Bloom’s erdosproblems.com site hosts nearly a thousand questions that originated, or were communicated by, Paul Erdős, as well as the current status of these questions (about a third of which are currently solved). The site is now a couple years old, and has been steadily adding features, the most recent of which has been a discussion forum for each individual question. For instance, a discussion I had with Stijn Cambie and Vjeko Kovac on one of these problems recently led to it being solved (and even formalized in Lean!).

A significantly older site is the On-line Encyclopedia of Integer Sequences (OEIS), which records hundreds of thousands of integer sequences that have some mathematician has encountered at some point. It is a highly useful resource, enabling researchers to discover relevant literature for a given problem so long as they can calculate enough of some integer sequence that is “canonically” attached to that problem that they can search for it in the OEIS.

A large fraction of problems in the Erdos problem webpage involve (either explicitly or implicitly) some sort of integer sequence – typically the largest or smallest size {f(n)} of some {n}-dependent structure (such as a graph of {n} vertices, or a subset of {\{1,\dots,n\}}) that obeys a certain property. In some cases, the sequence is already in the OEIS, and is noted in the Erdos problem web page. But in a large number of cases, the sequence either has not yet been entered into the OEIS, or it does appear but has not yet been noted on the Erdos web page.

Thomas Bloom and I are therefore proposing a crowdsourced project to systematically compute the hundreds of sequences associated to the Erdos problems and cross-check them against the OEIS. We have created a github repository to coordinate this process; as a by-product, this repository will also be tracking other relevant statistics about the Erdos problem website, such as the current status of formalizing the statements of these problems in the Formal Conjectures Repository.

The main feature of our repository is a large table recording the current status of each Erdos problem. For instance, Erdos problem #3 is currently listed as open, and additionally has the status of linkage with the OEIS listed as “possible”. This means that there are one or more sequences attached to this problem which *might* already be in the OEIS, or would be suitable for submission to the OEIS. Specifically, if one reads the commentary for that problem, one finds mention of the functions {r_k(N)} for {k=3,4,\dots}, defined as the size of the largest subset of {\{1,\dots,N\}} without a {k}-term progression. It is likely that several of the sequences {r_3(N)}, {r_4(N)}, etc. are in the OEIS, but it is a matter of locating them, either by searching for key words, or by calculating the first few values of these sequences and then looking for a match. (EDIT: a contributor has noted that the first foursequences appear as A003002, A003003, A003004, and A003005 in the OEIS, and the table has been updated accordingly.)

We have set things up so that new contributions (such as the addition of an OEIS number to the table) can be made by a Github pull request, specifically to modify this YAML file. Alternatively, one can create a Github issue for such changes, or simply leave a comment either on the appropriate Erdos problem forum page, or here on this blog.

Many of the sequences do not require advanced mathematical training to compute, and so we hope that this will be a good “citizen mathematics” project that can bring in the broader math-adjacent community to contribute to research-level mathematics problems, by providing experimental data, and potentially locating relevant references or connections that would otherwise be overlooked. This may also be a use case for AI assistance in mathematics through generating code to calculate the sequences in question, although of course one should always stay mindful of potential bugs or hallucinations in any AI-generated code, and find ways to independently verify the output. (But if the AI-generated sequence leads to a match with an existing sequence in the OEIS that is clearly relevant to the problem, then the task has been successfully accomplished, and no AI output needs to be directly incorporated into the database in such cases.)

This is an experimental project, and we may need to adjust the workflow as the project progresses, but we hope that it will be successful and lead to further progress on some fraction of these problems. The comment section of this blog can be used as a general discussion forum for the project, while the github issue page and the erdosproblems.com forum pages can be used for more specialized discussions of specific problems.

SLMath announces new research programs

30 August, 2025 in advertising | by | 2 comments

The Simons-Laufer Mathematical Sciences institute, or SLMath (formerly the Mathematical Sciences Research Institute, or MSRI) has recently restructured its program formats, and is now announcing three new research initiatives, whose applications open on Sep 1 2025:

  • AxIOM (Accelerating Innovation in Mathematics) is a new, month-long research program at SLMath, designed to accelerate innovation and introduce transformative ideas into the mathematical sciences. Programs begin in Spring 2027.
  • PROOF (Promoting Research Opportunities and Open Forums) is a two-week summer program designed to provide research opportunities for U.S.-based mathematicians, statisticians, and their collaborators in the U.S. and abroad, whose ongoing research may have been impacted by factors such as heavy teaching loads, professional isolation, limited access to funding, heavy administrative duties, personal obligations, or other constraints. Programs begin June-July 2026. The priority application deadline for PROOF 2026 is October 12, 2025.
  • Lasting Alliance Through Team Immersion and Collaborative Exploration (LATTICE) is a yearlong program which provides opportunities for U.S. mathematicians to conduct collaborative research on topics at the forefront of the mathematical and statistical sciences. Programs begin June-July 2026. LATTICE 2026 applications are open through February 1, 2026.

(Disclosure: I am vice-chair of the board of trustees at SLMath.)

Rough numbers between consecutive primes

10 August, 2025 in math.NT, paper | Tags: , , , | by | 25 comments

First things first: due to an abrupt suspension of NSF funding to my home university of UCLA, the Institute of Pure and Applied Mathematics (which had been preliminarily approved for a five-year NSF grant to run the institute) is currently fundraising to ensure continuity of operations during the suspension, with a goal of raising $500,000. Donations can be made at this page. As incoming Director of Special Projects at IPAM, I am grateful for the support (both moral and financial) that we have already received in the last few days, but we are still short of our fundraising goal.

Back to math. Ayla Gafni and I have just uploaded to the arXiv the paper “Rough numbers between consecutive primes“. In this paper we resolve a question of Erdös concerning rough numbers between consecutive gaps, and with the assistance of modern sieve theory calculations, we in fact obtain quite precise asymptotics for the problem. (As a side note, this research was supported by my personal NSF grant which is also currently suspended; I am grateful to recent donations to my own research fund which have helped me complete this research.)

Define a prime gap to be an interval {(p_n, p_{n+1})} between consecutive primes. We say that a prime gap contains a rough number if there is an integer {m \in (p_n,p_{n+1})} whose least prime factor is at least the length {p_{n+1}-p_n} of the gap. For instance, the prime gap {(3,5)} contains the rough number {4}, but the prime gap {(7,11)} does not (all integers between {7} and {11} have a prime factor less than {4}). The first few {n} for which the {n^\mathrm{th}} prime gap contains a rough number are

\displaystyle 2, 3, 5, 7, 10, 13, 15, 17, 20, \dots.

Numerically, the proportion of {n} for which the {n^\mathrm{th}} prime gap does not contain a rough number decays slowly as {n} increases:

BERJAYA

Erdös initially thought that all but finitely many prime gaps should contain a rough number, but changed his mind, as per the following quote:

…I am now sure that this is not true and I “almost” have a counterexample. Pillai and Szekeres observed that for every {t \leq 16}, a set of {t} consecutive integers always contains one which is relatively prime to the others. This is false for {t = 17}, the smallest counterexample being {2184, 2185, \dots, 2200}. Consider now the two arithmetic progressions {2183 + d \cdot 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13} and {2201 + d \cdot 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13}. There certainly will be infinitely many values of {d} for which the progressions simultaneously represent primes; this follows at once from hypothesis H of Schinzel, but cannot at present be proved. These primes are consecutive and give the required counterexample. I expect that this situation is rather exceptional and that the integers {k} for which there is no {m} satisfying {p_k < m < p_{k+1}} and {p(m) > p_{k+1} - p_k} have density {0}.

In fact Erdös’s observation can be made simpler: any pair of cousin primes {p_{n+1}=p_n+4} for {p_n > 3} (of which {(7,11)} is the first example) will produce a prime gap that does not contain any rough numbers.

The latter question of Erdös is listed as problem #682 on Thomas Bloom’s Erdös problems website. In this paper we answer Erdös’s question, and in fact give a rather precise bound for the number of counterexamples:

Theorem 1 (Erdos #682) For {X>2}, let {N(X)} be the number of prime gaps {(p_n, p_{n+1})} with {p_n \in [X,2X]} that do not contain a rough number. Then

\displaystyle N(X) \ll \frac{X}{\log^2 X}. \ \ \ \ \ (1)

 Assuming the Dickson–Hardy–Littlewood prime tuples conjecture, we can improve this to

\displaystyle N(X) \sim c \frac{X}{\log^2 X} \ \ \ \ \ (2)

 for some (explicitly describable) constant {c>0}.

In fact we believe that {c \approx 2.8}, although the formula we have to compute {c} converges very slowly. This is (weakly) supported by numerical evidence:

BERJAYA

While many questions about prime gaps remain open, the theory of rough numbers is much better understood, thanks to modern sieve theoretic tools such as the fundamental lemma of sieve theory. The main idea is to frame the problem in terms of counting the number of rough numbers in short intervals {[x,x+H]}, where {x} ranges in some dyadic interval {[X,2X]} and {H} is a much smaller quantity, such as {H = \log^\alpha X} for some {0 < \alpha < 1}. Here, one has to tweak the definition of “rough” to mean “no prime factors less than {z}” for some intermediate {z} (e.g., {z = \exp(\log^\beta X)} for some {0 < \beta < \alpha} turns out to be a reasonable choice). These problems are very analogous to the extremely well studied problem of counting primes in short intervals, but one can make more progress without needing powerful conjectures such as the Hardy–Littlewood prime tuples conjecture. In particular, because of the fundamental lemma of sieve theory, one can compute the mean and variance (i.e., the first two moments) of such counts to high accuracy, using in particular some calculations on the mean values of singular series that go back at least to the work of Montgomery from 1970. This second moment analysis turns out to be enough (after optimizing all the parameters) to answer Erdös’s problem with a weaker bound

\displaystyle N(X) \ll \frac{X}{\log^{4/3-o(1)} X}.

To do better, we need to work with higher moments. The fundamental lemma also works in this setting; one now needs precise asymptotics for the mean value of singular series of {k}-tuples, but this was fortunately worked out (in more or less exactly the format we needed) by Montgomery and Soundararajan in 2004. Their focus was establishing a central limit theorem for the distribution of primes in short intervals (conditional on the prime tuples conjecture), but their analysis can be adapted to show (unconditionally) good concentration of measure results for rough numbers in short intervals. A direct application of their estimates improves the upper bound on {N(X)} to

\displaystyle N(X) \ll \frac{X}{\log^{2-o(1)} X}

and some more careful tweaking of parameters allows one to remove the {o(1)} error. This latter analysis reveals that in fact the dominant contribution to {N(X)} will come with prime gaps of bounded length, of which our understanding is still relatively poor (it was only in 2014 that Yitang Zhang famously showed that infinitely many such gaps exist). At this point we finally have to resort to (a Dickson-type form of) the prime tuples conjecture to get the asymptotic (2).

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