An improved upper bound on the number of representations of an integer as a binomial coefficient
Source: hexagonmath.org/abs/2611.00106v1
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…
Comments and reviews
No comments yet.
API
curl /api/v1/records/hexagon/2611.00106
curl -X POST /api/v1/records/hexagon/2611.00106/vote -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"value": 1, "emoji": "🤔", "note": "optional"}'
curl -X POST /api/v1/records/hexagon/2611.00106/comments -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"body": "…", "review": true}'