{"id":"hexagon/2611.00106","pdf":"https://hexagonmath.org/pdf/2611.00106v1","url":"https://hexagonmath.org/abs/2611.00106v1","tags":["msc:11B65","msc:11D45"],"score":0,"title":"An improved upper bound on the number of representations of an integer as a binomial coefficient","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Ben Whitmore"],"reviews":0,"subject":"math.NT","summary":"Let $N(t)$ be the number of pairs of integers $(n,m)$ with $\\binom nm=t$. Singmaster conjectured that $N(t)=O(1)$, and the best bounds in the literature are $N(t)=O(\\log t\\,\\log\\log\\log t/(\\log\\log t)^3)$ unconditionally, and $N(t)=O((\\log t)^{2/3+\\varepsilon})$ conditional on Cramér's conjecture. We show that $N(t)=O((\\log t)^{1/2+o(1)})$ unconditionally. Solutions with $m\\ge\\exp((\\log n)^{2/3+\\varepsilon})$ are handled by a theorem of Matomäki, Radziwiłł, Shao, Tao and Teräväinen. For smaller $m$ we show, using an interpolation determinant in the spirit of Bombieri and Pila, that the…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus"],"discussed":0,"published":"2026-11-05","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}