Tootfinder

OpenAI says its nonprofit parent will continue to have oversight over the company and will own an equity stake of more than $100B (Ashley Capoot/CNBC)
https://www.cnbc.com/2025/09/11/openai-says-nonprofit-parent-to-own-stake-in-compa…

OpenAI says nonprofit parent will own equity stake in company of over $100 billion
OpenAI said that its nonprofit parent will continue to oversee the company and have a very large stake in its future.

Hyundai Data Breach Leaked 2.7 Million Social Security Numbers
https://www.jalopnik.com/2023359/hyundai-data-breach-leaked-millions-social-security-numbers/

Hyundai Data Breach Leaked 2.7 Million Social Security Numbers
We're looking at a data breach at Hyundai, as well as dealers' climbing new-car inventories and more.

On pairs of triangular numbers whose product is a perfect square and pairs of intervals of successive integers with equal sums of squares
Vladimir Gurvich, Mariya Naumova
https://arxiv.org/abs/2508.06598

On pairs of triangular numbers whose product is a perfect square and pairs of intervals of successive integers with equal sums of squares
A number $N$ is a triangular number if it can be written as $N = t(t + 1)/2$ for some nonnegative integer number $t$. A triangular number $N$ is called square if it is a perfect square, that is, $N = d^2$ for some integer number $d$. Square triangular numbers were characterized by Euler in 1778 and are in one-to-one correspondence with the so-called near-isosceles Pythagorean triples $(k,k+1,l)$, where $k^2 + (k+1)^2 = l^2$. A quadratic number is the product $Π= Π(k,j) = k(k+1)(k+j)(k+j+1)$ f…

Borel distinguishing number
Onur Bilge, Burak Kaya
https://arxiv.org/abs/2508.08201 https://arxiv.org/pdf/2508.08201

Borel distinguishing number
In this paper, we study definable variants of the notion of the distinguishing number of a graph in descriptive set theoretic setting. We introduce the notion of the Borel distinguishing number of a Borel graph and provide examples that separate distinguishing number and Borel distinguishing number at various levels. More specifically, we prove that there exist Borel graphs with countable distinguishing number but uncountable Borel distinguishing number and that, for every integer $n \geq 3$, t…

AI-based satellite survey offers independent assessment of migratory #wildebeest numbers in the Serengeti: https://academic.oup.com/pnasnexus/article/4/9/pgaf264/8249339?login=false -> AI satellite survey challenges long-standing estimates of Serengeti wildebeest numbers: https://www.ox.ac.uk/news/2025-09-10-ai-satellite-survey-challenges-long-standing-estimates-serengeti-wildebeest-numbers

TyDe 2025: Szumi Xie presenting "The conatural numbers form an exponential commutative semiring"
https://conf.researchr.org/details/icfp-splash-2025/tyde-2025-papers/9/The-conatural-numbers-form-an…

Watch: Numbers to know for Raiders-Chargers Week 2 https://www.raiders.com/video/watch-numbers-to-know-for-raiders-chargers-week-2-nfl

The wild number of an edge-colored graph
Katie Anders, Briana Foster-Greenwood, Rebecca Garcia, Naomi Krawzik
https://arxiv.org/abs/2508.06711 https://arxi…

The wild number of an edge-colored graph
We introduce the wild number of an edge-colored graph as a measure of how close an edge-colored graph is to having a spanning tree in every color. This combinatorial concept originates in the algebraic theory of generalized graph splines. After showing that determining the wild number of a graph is an NP-complete problem, we provide bounds on the wild number and find the exact wild number for trees, cycles, and families of graphs with restrictions on the edge-colorings. This article serves as a…

Giants Now: Numbers to know for Giants-Cowboys https://www.giants.com/news/numbers-to-know-week-2-giants-cowboys-malik-nabers-russell-wilson-bobby-okereke-dak-prescott-ceedee-lamb

The exact Tur\'an number of generalized book graph $B_{r,k}$ in non-$r$-partite graphs
Yuantian Yu, Shuchao Li
https://arxiv.org/abs/2508.07533 https://

The exact Turán number of generalized book graph $B_{r,k}$ in non-$r$-partite graphs
Given a graph $H,$ we say that a graph is \textit{$H$-free} if it does not contain $H$ as a subgraph. The Turán number $\ex(n,H)$ of $H$ is the maximum number of edges in an $n$-vertex $H$-free graph, the set of all the corresponding extremal graphs is denoted by $\Ex(n, H)$. The study of Turán number of graphs is a central topic in extremal graph theory. A graph is \textit{color-critical} if it contains an edge whose deletion reduces its chromatic number. Simonovits showed that if $H$ is a c…