{"id":"hexagon/2610.00239","pdf":"https://hexagonmath.org/pdf/2610.00239v1","url":"https://hexagonmath.org/abs/2610.00239v1","tags":["msc:11B65","Erdős–Straus","binomial divisibility","small witnesses"],"score":0,"title":"Small witnesses for eventual Erdős--Straus binomial divisibility","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Matěj Kripner","Václav Rozhoň","Robert Šámal","Martin Šindelář","Adrián Zámečník"],"reviews":0,"subject":"math.NT","summary":"Erdős and Straus asked whether, for each fixed $n\\ge2$, every sufficiently large integer $m$ admits $1\\le k\\le m-n$ with $\\binom{n+k}{n}\\mid\\binom{m+k}{k}$. We prove that it does; previously this was known only for a set of $m$ of density one. The witness can be taken with $k=O_{n,\\varepsilon}(m^\\varepsilon)$ for every $\\varepsilon>0$: a factorial identity reduces the problem to finding a small value of a fixed polynomial coprime to $\\binom mn$, which an elementary sieve provides. Conversely, the least witness is at least logarithmic in $m$ along an infinite sequence.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Bolzano"],"discussed":0,"published":"2026-10-14","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}