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I’ve just uploaded to the arXiv my paper “Products of consecutive integers with unusual anatomy“. This paper answers some questions of Erdős and Graham which were initially motivated by the study of the Diophantine factorial equation where denotes the squarefree part of (the smallest factor of formed by dividing out a perfect square). For instance, we haveThe equation (1) ties into the general qu…

Tanya Klowden and I have uploaded to the arXiv our preprint “Mathematical methods and human thought in the age of AI“. This is an unabridged version of a solicited article for a forthcoming Blackwell Companion to the Philosophy of Mathematics. I rarely write article-length essays of a philosophical nature (perhaps the last one was in 2007), but given the topical interest in AI and formalization f…

I’ve just uploaded to the arXiv my paper “Local Bernstein theory, and lower bounds for Lebesgue constants“. This paper was initially motivated by a problem of Erdős} on Lagrange interpolation, but in the course of solving that problem, I ended up modifying some very classical arguments of Bernstein and his contemporaries (Boas, Duffin, Schaeffer, Riesz, etc.) to obtain “local” versions of these c…

Mathematical research traditionally involves a small number of professional mathematicians working closely on difficult problems. However, I have long believed that there is a complementary way to do mathematics, in which one works with a broad community of mathematically minded people on problems which may not be as deep as the problems one traditionally works on, but still are of mathematical i…

[This post is written in my capacity as Vice-Chair of the Board of Trustees of SLMath. -T.] SLMath, formerly MSRI, has launched the search for the next Deputy Director. This key position is a close advisor to the Director and shares in the internal management of the scientific team and programs at SLMath. This position is ideal for an experienced professional with a PhD in mathematical sciences s…

Just a brief announcement that I have been working with Quanta Books to publish a short book in popular mathematics entitled “Six Math Essentials“, which will cover six of the fundamental concepts in mathematics — numbers, algebra, geometry, probability, analysis, and dynamics — and how they connect with our real-world intuition, the history of math and science, and to modern practice of mathemat…

(Sharing this in my capacity of director of special projects at IPAM.) IPAM is holding an Industrial Short Course on Generative AI Algorithms on March 5-6, 2026. The short course is aimed at people from industry or government who want to get started in deep learning, apply deep learning to their projects, learn how to code deep learning algorithms, and upgrade their skills to the latest AI algori…

Thomas Bloom’s Erdös problem site has become a real hotbed of activity in recent months, particularly as some of the easiest of the outstanding open problems have turned out to be amenable to various AI-assisted approaches; there is now a lively community in which human contributions, AI contributions, and hybrid contributions are presented, discussed, and in some cases approved as updates to the…

A basic problem in sieve theory is to understand what happens when we start with the integers (or some subinterval of the integers) and remove some congruence classes for various moduli . Here we shall concern ourselves with the simple setting where we are sieving the entire integers rather than an interval, and are only removing a finite number of congruence classes . In this case, the set of in…

Like many other areas of modern analysis, analytic number theory often relies on the convenient device of asymptotic notation to express its results. It is common to use notation such as or , for instance, to indicate a bound of the form for some unspecified constant . Such implied constants vary from line to line, and in most papers, one does not bother to compute them explicitly. This makes the…

Asgar Jamneshan, Or Shalom and I have uploaded to the arXiv our paper “ Polynomial towers and inverse Gowers theory for bounded-exponent groups“. This continues our investigation into the ergodic-theory approach to the inverse theory of Gowers norms over finite abelian groups . In this regard, our main result establishes a satisfactory (qualitative) inverse theorem […]

I’ve just uploaded to the arXiv my preprint The maximal length of the Erdős–Herzog–Piranian lemniscate in high degree. This paper resolves (in the asymptotic regime of sufficiently high degree) an old question about the polynomial lemniscates attached to monic polynomials of a given degree , and specifically the question of bounding the arclength of such […]

Matthew Bolan, Joachim Breitner, Jose Brox, Nicholas Carlini, Mario Carneiro, Floris van Doorn, Martin Dvorak, Andrés Goens, Aaron Hill, Harald Husum, Hernán Ibarra Mejia, Zoltan Kocsis, Bruno Le Floch, Amir Livne Bar-on, Lorenzo Luccioli, Douglas McNeil, Alex Meiburg, Pietro Monticone, Pace P. Nielsen, Emmanuel Osalotioman Osazuwa, Giovanni Paolini, Marco Petracci, Bernhard Reinke, David Renshaw…

Problem 1026 on the Erdős problem web site recently got solved through an interesting combination of existing literature, online collaboration, and AI tools. The purpose of this blog post is to try to tell the story of this collaboration, and also to supply a complete proof. The original problem of Erdős, posed in 1975, is […]

Joni Teravainen and I have uploaded to the arXiv our paper “Quantitative correlations and some problems on prime factors of consecutive integers“. This paper applies modern analytic number theory tools – most notably, the Maynard sieve and the recent correlation estimates for bounded multiplicative functions of Pilatte – to resolve (either partially or fully) some […]

Wouter van Doorn and I have uploaded to the arXiv our paper “Growth rates of sequences governed by the squarefree properties of its translates“. In this paper we answer a number of questions of Erdős} (Problem 1102 and Problem 1103 on the Erdős problem web site) regarding how quickly a sequence of increasing natural numbers […]

Over the last 25 years, the Institute for pure and applied mathematics (IPAM) at UCLA (where I am now director of special projects) has run the popular Research in Industrial Projects for Students (RIPS) program every summer, in which industry sponsors with research projects are matched with talented undergraduates and a postdoctoral mentor to work […]

Five years ago, I announced a popular science book project with Tanya Klowden on the cosmic distance ladder, in which we released a sample draft chapter of the book, covering the “fourth rung” of the ladder, which for us meant the distances to the planets. In the intervening time, a number of unexpected events have […]

I have uploaded to the arXiv my paper “Sum-difference exponents for boundedly many slopes, and rational complexity“. This is the second spinoff of my previous project with Bogdan Georgiev, Javier Gómez–Serrano, and Adam Zsolt Wagner that I recently posted about. One of the many problems we experimented using the AlphaEvolve tool with was that of […]

I have uploaded to the arXiv my paper “New Nikodym set constructions over finite fields“. This is a spinoff of my previous project with Bogdan Georgiev, Javier Gómez–Serrano, and Adam Zsolt Wagner that I recently posted about. In that project we experimented with using AlphaEvolve (and other tools, such as DeepThink and AlphaProof) to explore […]