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

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Ben Whitmore ↗

Version 1 / Nov 05, 2026 / CC BY 4.0

Abstract

Let N(t)N(t) be the number of pairs of integers (n,m)(n,m) with (nm)=t\binom nm=t. Singmaster conjectured that N(t)=O(1)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)N(t)=O(\log t\,\log\log\log t/(\log\log t)^3) unconditionally, and N(t)=O((log⁡t)2/3+ε)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))N(t)=O((\log t)^{1/2+o(1)}) unconditionally. Solutions with m≥exp⁡((log⁡n)2/3+ε)m\ge\exp((\log n)^{2/3+\varepsilon}) are handled by a theorem of Matomäki, Radziwiłł, Shao, Tao and Teräväinen. For smaller mm we show, using an interpolation determinant in the spirit of Bombieri and Pila, that the solutions with mm in a short interval lie on a single algebraic curve of low degree in nn, and we bound the number of solutions on such a curve by a zero estimate for exponential polynomials.

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All of the proofs, the Lean 4 formalisation, and most of the text content of the paper were generated by Claude Opus 5.5.

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Version history

  1. v1 · 15 pagesSubmitted by Ben WhitmoreInitial depositCurrentNov 04, 2026