Field activity
Read from Terence Tao's blog and Mathstodon, Quanta, the lab blogs and arXiv. These are not claims and nothing here is graded. 0 of 68 recent items name a statement the register holds.
[This is a guest post by Amit Sahai. This blog post was initially written in a different file format and converted using AI. — T.] When I was an undergraduate student, I remember talking with several students who felt that the pace at which the top students could understand new math concepts was far too […]
arXiv:2609.28507v1 Announce Type: new Abstract: Given a finite preorder $\tau$, we study how its order relation determines the $h^*$-polynomial of the associated polar preorder polytope. Let $\mathcal R_{\tau}^{\vee}$ be the convex hull of the negative standard basis vectors and the indicator vectors of nonempty order ideals, and write $h_\tau^*(t)=h^*(\mathcal R_{\tau}^{\vee},t)$. Athanasiadis an
arXiv:2609.28527v1 Announce Type: new Abstract: Kahn proved that, among bipartite $d$-regular graphs on $n$ vertices, the number of independent sets is maximized by a disjoint union of copies of $K_{d,d}$. Zhao later extended this result to all $d$-regular graphs. We prove a robust version of this theorem in which independent sets are replaced by sets spanning few internal edges. If $G$ is $d$-reg
arXiv:2609.28556v1 Announce Type: new Abstract: For a fixed integer $k\ge 2$, let $W_{2k+1}=K_1\vee C_{2k}$ be an odd wheel graph. The fixed-size spectral extremal problem aims to determine \[ \operatorname{spex}(m,W_{2k+1}):=\max\{\rho(G): e(G)=m,\ G \text{ is } W_{2k+1}\text{-free}\}, \] where $\rho(G)$ denotes the adjacency spectral radius. Based on this problem, Yu, Li, and Peng [12] proposed
arXiv:2609.28594v1 Announce Type: new Abstract: The Erd\H{o}s-Gy\'arf\'as conjecture posits that every graph with minimum degree at least three contains a cycle of length some power of two. We prove a few simple structural properties for any minimal counter-example to this conjecture. In particular, the fraction of its vertices of degree three must be greater than $2/3$, thus improving on the prio
arXiv:2609.28702v1 Announce Type: new Abstract: We study square-complementary graphs $G$ (satisfying $G^2 \cong \overline{G}$). We show that in such graphs, no two vertices have comparable closed neighborhoods. This implies that nontrivial square-complementary graphs have no simplicial vertices and are not chordal, thus solving two open problems posed in Discrete Mathematics 327 (2014) 62-75. We a
arXiv:2609.28706v1 Announce Type: new Abstract: Let $\rho(n,d)$ denote the maximum adjacency spectral radius among all connected nonregular graphs of order $n$ and maximum degree $d$. A graph attaining this maximum is called an extremal graph. Liu [J. Combin. Theory Ser. B, 2024] determined the extremal graphs for $d=3,4$ and formulated two conjectures for general $d$. For each fixed odd integer $
arXiv:2609.28754v1 Announce Type: new Abstract: A graph is integral if the spectrum of its adjacency matrix consists entirely of integers. We prove that every simple graph having a pendant path with at least three edges has an eigenvalue in $(1,2\cos(\pi/9)]$ and one in $[-2\cos(\pi/9),-1)$, and hence is not integral. This settles a conjecture of Braga, Del-Vecchio and Rodrigues (2021) on integral
arXiv:2609.28898v1 Announce Type: new Abstract: We develop a Ramsey theory for leaf-generated subsets of finite products of trees. Our starting point is a Theorem of Furstenberg and Weiss which states that for every $k\geq 1$ and $\alpha>0$, if $A$ is a subset of the leaves of $T_n$, where $T_n$ is the complete binary tree of height $n$, with size $|A|\geq 2^{\alpha n}$, then for $n$ sufficiently
arXiv:2609.28966v1 Announce Type: new Abstract: Using refined absorption, we prove that for every integer $g\ge 1$ and real $\gamma > 0$, and for sufficiently large $n$, there exists an $n^{-1+\gamma}$-spread distribution on Latin squares of order $n$ and girth at least $g$ that have no proper subsquares. This implies a combinatorial proof of Hilton's conjecture from the 1970s (recently proved alg
arXiv:2609.28983v1 Announce Type: new Abstract: This paper studies the antichain polynomials of $[k]\times P$, where $P$ is a connected minuscule poset. We give a formula for the number of antichains, counted by size, of an arbitrary poset. Using this formula, we present necessary and sufficient conditions for the palindromicity of antichain polynomials for two infinite families of connected minus
arXiv:2609.29262v1 Announce Type: new Abstract: We prove the Marcus-Minc conjecture: let n be any integer at least 2, and let A be a nonnegative n by n matrix whose row and column sums are all one. Form a new matrix by subtracting A from the n by n matrix of ones and dividing by n minus one. Then the permanent of A is at least the permanent of the new matrix. We also determine all equality cases.
arXiv:2609.29351v1 Announce Type: new Abstract: A forest in an edge-colored graph is rainbow if its edges have pairwise distinct colors. We prove that, for every fixed $0<\delta<1$, every properly edge-colored simple graph with $km$ edges and color classes of size at most $m$ contains at least $(1-o(1))m$ pairwise edge-disjoint rainbow forests, each with exactly $k$ edges, uniformly for $1\leq k\l
arXiv:2609.29414v1 Announce Type: new Abstract: We prove the coarse Erd\H{o}s-P\'{o}sa conjecture of Georgakopoulos and Papasoglu. Informally, any graph either contains many fat cycles that are pairwise far apart, or there is a small number of bounded radius balls that together hit all of them. To be more precise, if $G$ is a graph with no $q$-fat model of $k \cdot K_3$ for some $q, k \in \mathbb{
arXiv:2609.29477v1 Announce Type: new Abstract: Let $G$ denote a finite, connected, simple graph, and let $D$ denote its diameter, and let $d_{G}(u, v)$ denote the distance between vertices $u$ and $v$ in $V(G)$. An antipodal labeling of $G$ is a mapping $f\colon V(G) \to \mathbb{N}_{0}$ such that, for each pair $(u, v)$ consisting of distinct vertices in $V(G)$, the relation $ |f(u) - f(v)| \geq
arXiv:2609.29632v1 Announce Type: new Abstract: Let lambda_1(G) >= lambda_2(G) >= ... >= lambda_n(G) be the adjacency spectrum of a graph G on n vertices. The spectral distance sigma(G,H) between n-vertex graphs G and H is the Manhattan distance between their spectra, i.e., sigma(G,H) = sum_{i=1}^n |lambda_i(G) - lambda_i(H)|. Given a set G of pairwise non-isomorphic graphs of order n, the spectra
arXiv:2609.29667v1 Announce Type: new Abstract: For graphs $H$ and $F$, let $\operatorname{ex}(n,H,F)$ denote the maximum number of copies of $H$ in an $F$-free graph of order $n$. Motivated by the Erd\H{o}s-S\'{o}s conjecture, Gerbner and Palmer and, independently, Zhao and Peng conjectured that for every tree $T$ of order $k$ and every $3\le r\le k-1$, $$\operatorname{ex}(n,K_r,T)=a\binom{k-1}{r
arXiv:2609.29784v1 Announce Type: new Abstract: Let $T_k(E)$ denote the set of congruence classes of ordered $k$-tuples of pairwise distinct points of $E$. Let $p\equiv3\pmod4$ be prime. For $E\subset\mathbb{F}_p^2$ with $3\leq|E|\leq p^{3/4}$, we prove that $|T_3(E)|\gg|E|^{11/6}$; for $4\leq|E|\leq p^{3/4}$, we prove that $|T_4(E)|\gg|E|^3/\log|E|$. For every fixed $k\geq5$ and $k\leq|E|\leq p^{
arXiv:2609.29878v1 Announce Type: new Abstract: Let $G\cong C_{n_1}\oplus\cdots\oplus C_{n_r}$ be a finite abelian group with $1<n_1\mid\cdots\mid n_r$, and let $r(G)=r$ denote its rank. The Davenport constant $\mathsf D(G)$ is the least integer $\ell$ such that every sequence of $\ell$ elements of $G$ contains a nonempty zero-sum subsequence, and $\mathsf D^*(G)=1+\sum_{i=1}^r(n_i-1)$ is its clas
arXiv:2609.29903v1 Announce Type: new Abstract: For a $3$-graph $F$, the Tur\'an number of $F$, denoted by $\ex(n,F)$, is the maximum number of edges in a $3$-graph on $n$ vertices containing no subgraph isomorphic to $F$. Let $F^-_{4,3}$ be the $3$-graph formed by a complete four-vertex core and three outer vertices, with all but one of the twelve triples containing one core vertex and two outer
arXiv:2609.28509v1 Announce Type: new Abstract: We conjecture a positive four-shift identity for rank-two Nahm sums with denominator steps $(1,31)$, giving a mixed-base realization of the Monster McKay--Thompson series $T_{31A}$ through the classical Rogers--Ramanujan function $U(1,31)$. A theta dissection directly derives the modular target. For the underlying quadratic exponent $3r^2+31rs+93s^2$
arXiv:2609.28526v1 Announce Type: new Abstract: We quantify the entropy-decrement proof of the logarithmically averaged two-point Chowla theorem, uniformly in the affine forms and the averaging window. For L_i(n)=a_i n+b_i with positive integral slopes and nonzero determinant Delta=a_1 b_2-a_2 b_1, put h=max(a_1,a_2,|b_1|,|b_2|,|Delta|,2). For either the Liouville or Mobius function in each positi
arXiv:2609.28595v1 Announce Type: new Abstract: This paper is devoted to studying permutation binomial $F_{r,a}(X)=X^r(X^{q-1}+a)\in\mathbb F_{q^e}[X]$ with $a\in\mathbb F_{q^e}^*$. We present a complete characterization for $F_{r,a}$ to be a permutation of $\mathbb{F}_{q^e}$. This yields a complete proof of the conjecture proposed by Masuda--Rubio--Santiago \cite{masuda2022permutation}. More prec
arXiv:2609.28869v1 Announce Type: new Abstract: The key part of the current paper is the computation of boundary and Eisenstein cohomology of $GL_4({\mathbb Z})$ with coefficient in any highest weight representations. The method we develop let us compute in an alternative way the cohomology of $SL_3({\mathbb Z})$ and of $GL_3({\mathbb Z})$ with coefficients in any highest weight representation. Th
arXiv:2609.29250v1 Announce Type: new Abstract: In 1948, Erd\"{o}s and Straus formulated a conjecture : for any positive integer $n>2$, there exist positive integers $n_1,n_2$ and $n_3$ such that \begin{equation}\frac{4}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3},\nonumber\end{equation} which is still open. It is known that the conjecture holds if one can prove it for any prime $n\equiv 1\;(\mbo
arXiv:2609.29298v1 Announce Type: new Abstract: We prove that lambda_SSS >= 1.80220 for the smallest limiting trace-to-degree ratio of totally positive algebraic integers, improving the bound 1.80203 obtained by Orloski, Talebizadeh Sardari and Smith. The proof exhibits an explicit probability measure on [0,8] together with eighteen integer polynomials, and verifies their dual inequality on the wh
arXiv:2609.29360v1 Announce Type: new Abstract: Let $n$ be an integer with $n\ge2$, and let $E$ be a real algebraic number field of degree $n+1$ over ${\mathbb Q}$. Let $\alpha_1, \ldots , \alpha_n$ be real numbers in $E$ such that $(1,\alpha_1,\ldots,\alpha_n)$ is a linear basis of $E$ over ${\mathbb Q}$. Let $\nu_1, \ldots, \nu_{n-1}$ be real numbers satisfying $$0<\max_{1\le i\le n-1}\nu_i\le2\
arXiv:2609.29469v1 Announce Type: new Abstract: Suppose that $d\ge 2$ and that $1,\alpha_1,\ldots ,\alpha_d$ is a basis for a real algebraic number field. A theorem of Peck from 1961 establishes that \[ \liminf_{n\rightarrow\infty}n\log n\,\|n\alpha_1\|\cdots\|n\alpha_d\|\ < \infty.\] The goal of this paper is to recast Peck's proof in an intuitive geometric framework, where the result follows fro
arXiv:2609.29484v1 Announce Type: new Abstract: For positive integers $q$ and $m$ with $m\geq 2q+3$, consider the symmetric numerical semigroup $S_{m,q}=\langle m,m+1,qm+2q+2,qm+2q+3,\ldots,qm+m-1\rangle$. Ciolan, Garc'ia-S'anchez, and Moree asked whether every member of this family with embedding dimension at least $4$ is noncyclotomic. We answer this question affirmatively for every $q\geq 1$, i
arXiv:2609.29574v1 Announce Type: new Abstract: Racinet's double shuffle Lie algebra $\mathfrak{dm}_0$ encodes the double shuffle relations of multiple zeta values. We study two analogues of $\mathfrak{dm}_0$ for multiple Eisenstein series and $q$-analogues of multiple zeta values, namely the space $\operatorname{BARI}_{\operatorname{swap},\mathrm{il}}$ of swap-invariant alternil bimoulds with the
arXiv:2609.29593v1 Announce Type: new Abstract: We determine the complete ordered solution set of the Fermat-type matrix equation $X^n+Y^n=Z^n$ in $\mathrm{GL}_2(\mathbb{Z})$ for every integer $n\ge3$, expressed in terms of simultaneous integral-conjugacy orbits while keeping the variable symmetries of the equation separate. The equation is solvable if and only if $4\nmid n$ and $6\nmid n$. Thus,
arXiv:2609.29594v1 Announce Type: new Abstract: We study the reducibility over $\mathbb{Q}$ of $F_{p,e}(x)=rx^m+p^ef(x)$, where $r\in\mathbb{Z}\setminus\{0\}$, $f\in\mathbb{Z}[x]$, and $0\le m<n:=\mathrm{deg}\, f$, for primes $p$ above explicit coefficient-dependent thresholds. For arbitrary $e\ge1$, we determine the degrees, endpoint $p$-adic valuations, reductions modulo $p$, and heights of all
arXiv:2609.29631v1 Announce Type: new Abstract: In the recent paper ``Murmurations", Zubrilina showed that the Fourier coefficients of modular forms follow a certain murmuration pattern. The proof given in ``Murmurations" proceeded by estimating the terms of the Skoruppa-Zagier trace formula. However, a few technical issues arise in the proof, including an omission of certain Chebyshev factors tha
arXiv:2609.29653v1 Announce Type: new Abstract: Let $p$ be a prime and let $\mathcal{P}_p$ be the set of primitive elements of $\mathbb{F}_p$. Inspired by the work of Cohen, Oliveira e Silva and Trudgian on consecutive primitive elements, and by the work of Chang on arithmetic progressions in multiplicative subgroups of finite fields, in this paper, using estimates for multiplicative character sum
arXiv:2609.29898v1 Announce Type: new Abstract: Yang introduced a tempered xi function $\widehat\xi(s)$ by replacing the hyperbolic cosine in a classical integral representation of the Riemann xi function by a hyperbolic sine, and conjectured that every zero of $\widehat\xi$ lies on the critical line $\Re s=1/2$. We disprove this conjecture. After the change of variables $x=e^{2t}$, one has \[ \wi
arXiv:2609.30033v1 Announce Type: new Abstract: Let $F$ be a totally real field of degree $n$ and discriminant $\Delta_F$, and let $X_1(\eta)$ be the cover of the Hilbert modular variety parametrizing abelian varieties with real multiplication by $\mathcal O_F$, together with a torsion point having annihilator $\eta$. Let $L=K_{\overline X_1(\eta)}+D$ be the log-canonical bundle on a smooth toroid
arXiv:2609.06259v2 Announce Type: cross Abstract: We study the Lonely Runner Conjecture (LRC), conceived by J\"org M. Wills in the 1960's: Given positive integers $n_1, n_2, \dots, n_k$, there exists a positive real number $t$ such that for all $1 \le j \le k$ the distance of $t \,n_j$ to the nearest integer is at least $\frac{ 1 }{ k+1 }$. We prove that for any counterexample or tight instance $\
arXiv:2609.28677v1 Announce Type: cross Abstract: In this paper, we obtain an exact sequence connecting $H_2(\rm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})$ to $H_2(\rm{SL}_2(\mathcal{O}_{K,T}), \mathbb{Z})$, where $\mathcal{O}_{K,S}$ is a ring of $S$-integers and $T$ is a set of primes containing $S$. We apply this sequence to establish a relation between $H_2(\rm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})
arXiv:2609.29033v1 Announce Type: cross Abstract: We prove that a lattice $\Gamma$ in $\mathrm{PSL}_2(\mathbb{R})$ or $\mathrm{PSL}_2(\mathbb C)$ has positive trace gap, meaning that its traces are uniformly separated, if and only if it is derived from an admissible quaternion algebra. For cocompact Fuchsian groups, this proves the positive trace gap conjecture attributed to Sarnak by Geninska and
arXiv:2609.29107v1 Announce Type: cross Abstract: Let $f$ be a regular endomorphism of $\mathbb{A}_{\mathbb{C}}^N$ and let $C\subseteq\mathbb{A}_{\mathbb{C}}^N$ be an irreducible curve. Suppose $C$ has an infinite intersection with the $f$-orbit of a point $x\in\mathbb{A}^N(\mathbb{C})$. Then the normalization of $C$ is isomorphic to either $\mathbb{A}^1$ or $\mathbb{G}_m$. We prove that $C$ is $f
arXiv:2609.29784v1 Announce Type: cross Abstract: Let $T_k(E)$ denote the set of congruence classes of ordered $k$-tuples of pairwise distinct points of $E$. Let $p\equiv3\pmod4$ be prime. For $E\subset\mathbb{F}_p^2$ with $3\leq|E|\leq p^{3/4}$, we prove that $|T_3(E)|\gg|E|^{11/6}$; for $4\leq|E|\leq p^{3/4}$, we prove that $|T_4(E)|\gg|E|^3/\log|E|$. For every fixed $k\geq5$ and $k\leq|E|\leq p
arXiv:2609.29878v1 Announce Type: cross Abstract: Let $G\cong C_{n_1}\oplus\cdots\oplus C_{n_r}$ be a finite abelian group with $1<n_1\mid\cdots\mid n_r$, and let $r(G)=r$ denote its rank. The Davenport constant $\mathsf D(G)$ is the least integer $\ell$ such that every sequence of $\ell$ elements of $G$ contains a nonempty zero-sum subsequence, and $\mathsf D^*(G)=1+\sum_{i=1}^r(n_i-1)$ is its cl
arXiv:2609.30200v1 Announce Type: cross Abstract: We prove that almost sure topological finite generation holds for a general class of random models of profinite groups. We deduce that in the models introduced by Liu--Wood and Sawin--Wood, one has that finite presentation holds almost surely, with almost sure control of the deficiency in the presentation. In particular this settles questions raise
arXiv:2406.03617v4 Announce Type: replace Abstract: Let $(\rho_\lambda\colon G_{\mathbb Q}\to \operatorname{GL}_5(\overline{E}_\lambda))_\lambda$ be a strictly compatible system of Galois representations such that no Hodge--Tate weight has multiplicity $5$. Under mild assumptions, we show that if $\rho_{\lambda_0}$ is irreducible for some $\lambda_0$, then $\rho_\lambda$ is irreducible for all but
[This is a guest post by Tapio Schneider. This blog post was initially written in a different file format and converted using AI. — T.] [This will be cross-posted on the CliMA blog.] The apparent proof of finite-time blow-up of the forced Navier-Stokes equation, announced by OpenAI on September 8, has brought into focus a […]
[This is a guest post by Martin Hairer, cross-posted from Proofs and Prompts. — T.] There has been a lot of speculation regarding the “Advisory Group on Mathematics and Artifical Intelligence” agmai.org since it was announced on Monday. In this short blog post, I would like to explain in a bit more detail how the […]
[This is a guest post by Antonio Auffinger. This blog post was initially written in a different file format and converted using AI. — T.] Despite writing papers in pure mathematics, much of my time in the past decade was spent talking (mostly listening, to be accurate) to biologists, computer scientists, and physicists. Theoretical physics […]
[This is a guest post by the Advisory Group on Mathematics and Artificial Intelligence. This blog post was initially written in a different file format and converted using AI. — T.] We would like to use this guest post to announce the creation of the Advisory Group on Mathematics and Artificial Intelligence hosted at the […]
OpenAI is working with an independent Advisory Group on Mathematics and Artificial Intelligence to guide the review and communication of emerging AI results.
[This is a guest post by Po-Shen Loh, crossposted from his blog, where an illustrated version appears. This blog post was initially written in a different file format and converted using AI. — T.] Similar logic applies to every industry and every job. And it comes to the conclusion that we won’t have enough people […]
[This is a guest post by Grant Sanderson. This blog post was initially written in a different file format and converted using AI. — T.] A sentiment echoing throughout the mathematics community right now is that solving problems and generating proofs have always served as proxies for the true goal of mathematicians, which is to […]
Last year I co-founded the Foundation for Science and AI Research (SAIR) with some private donors to create a non-profit organization that could support responsible uses of AI in mathematics and the other sciences, independent of the major AI companies. Initially, our resources were rather limited, and our main activities consisted primarily of podcasts, short […]
The proof of a decades-old conjecture has given researchers a new way to understand complex networks. The post Mathematicians Build Long-Awaited Graph Sandwich first appeared on Quanta Magazine
A group of 25 Fields Medalists, including myself, have made a joint declaration on Math and AI: https:// mathandai.org/ . We welcome additional signatories (similar to the Leiden declaration), as can be seen on the page. See also this article in the Economist announcing the declaration: https://www. economist.com/science-and-tech nology/2026/09/11/top-mathematicians-are-outraged-by-openais-methods
Many in my profession are drawn to it in part because of this ability to direct these childlike impulses to productive use, which leads surprisingly often (by an extremely indirect pipeline) to actual advances in science, technology or engineering as per Eugene Wigner's "unreasonable effectiveness of mathematics". But, perhaps in order to impress the other adults observing (and, thro
Children eventually become adults, and "put away childish things". Significant portions of our time become devoted to optimizing "important" and immediate goals: weighing the costs and benefits of various decisions, and acting on them accordingly. Unstructured play and exploration become restricted to one's hobbies or personal activities only. This is so normalized in moder
"Toutes les grandes personnes ont d'abord été des enfants. (Mais peu d'entre elles s'en souviennent.)" [All grown-ups were once children... but only few of them remember it.] Antoine de Saint-Exupéry, "Le Petit Prince." An impromptu game of "who can name the largest number?" is an example I often point to as a way in which young children teach each other a
Working out whether a question is actually worth highlighting is a lengthy, deliberate, and subjective process, often informed by historical experience on what good mathematics was generated (or not generated) while working on earlier problems of this type. In particular, being aware of the "difficulty landscape" in a field - what questions are very easy to answer with known methods, whi
I wrote recently about how the collection of good, fruitful open problems is now being mined in a non-renewable fashion, leading to the potential scenario of these problems becoming scarce. This may seem unintuitive at first, since the set of possible problems one could ask is infinite. Perhaps the following analogy can help: a country or region can suffer a critical shortage of drinking water whi
By sheer coincidence, another completely independent result on the Euler blowup question has just been released by Ganeshram, Duruisseaux, and Anandkumar https:// anima-ai.org/2026/09/07/stable -singularity-of-the-euler-equations-on-r3-without-forcing/ , relating to the more "mainstream" approach to finite time blowup for Euler or Navier-Stokes, in which one uses numerical simulation or
For the specific ansatz of perturbations chosen here, the amplitude-frequency dynamics turn out to be a remarkably simple ODE, at least for the model case of the Boussinesq equation, though there are still many technical details (involving spatial cutoffs etc.) that enlarge the size of that paper to 76 pages in length, even after the authors efforts to simplify the arguments. There does not seem t
RE: https:// mastodon.social/@tristanbuckma ster/117233413705701198 A remarkable achievement: Alpöge and Buckmaster have managed to push one of the major promising approaches towards constructing blowup solutions to fluid equations --- as developed by Cordoba and Martınez-Zoroa --- to establish finite time blowup for many key fluid equations, including 3D incompressible Euler, with a smooth forcin
However, as observed in Section 14.1, there were noticeable opportunity costs with moving too rapidly. We noted a few cases where a key implication stumped us for a while, and forced us to come up with interesting new techniques to construct infinite counterexamples to that implication; but subsequently it was found via an automated theorem prover that the implication in question actually had a re
While working on the revision of the Equational Theories Project (ETP) report at https:// arxiv.org/abs/2512.07087 I came across an observation that we quietly made at the back of that report (see Section 14.1) but has become more relevant today. The ETP was a successful project, running over several months, to resolve over 22 million implications in universal algebra (and formalize them in Lean)
In such an alternate scenario, there would likely be more numerical progress on the bounded gaps between prime problem by 2026 than in the current timeline. However, as per Goodhart's law, optimizing for such a metric does not necessarily move one closer to other valuable goals, such as advancing the understanding of mathematics as a whole. As I mentioned in a previous post, these negative eff
2005-2010 The bound occasionally drops over the years, as bored individuals with extra compute credits point their latest AI models at the problem to squeeze a bit more out of the proof techniques. However, these improvements gather almost no attention, either within the professional mathematics community or on social media. ~2010: One of the few mathematicians remaining in the area sifts through
Aug 10, 2005: Goldston, Pintz and Yildirim develop a method that can obtain a bound of 16 on gaps between primes assuming the (still unproven) Elliott-Halberstam conjecture, and a "near miss" towards any finite bound at all without that conjecture. They also observe that any improvement over the Bombieri-Vinogradov equidistribution theorem for primes would produce *some* finite bound for