Ben Whitmore

math.NT — Number Theory

An improved upper bound on the number of representations of an integer as a binomial coefficient

Contributed by Ben Whitmore

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 solutions with $m$ in a short interval lie on a single algebraic curve of low degree in $n$, and we bound the number of solutions on such a curve by a zero estimate for exponential polynomials.