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Small witnesses for eventual Erdős--Straus binomial divisibility

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

Contributed by Matěj Kripner ↗, Václav Rozhoň ↗, Robert Šámal ↗, Martin Šindelář, Adrián Zámečník

Version 1 / Oct 14, 2026 / CC BY 4.0

Abstract

Erdős and Straus asked whether, for each fixed n≥2n\ge2, every sufficiently large integer mm admits 1≤k≤m−n1\le k\le m-n with (n+kn)∣(m+kk)\binom{n+k}{n}\mid\binom{m+k}{k}. We prove that it does; previously this was known only for a set of mm of density one. The witness can be taken with k=On,ε(mε)k=O_{n,\varepsilon}(m^\varepsilon) for every ε>0\varepsilon>0: a factorial identity reduces the problem to finding a small value of a fixed polynomial coprime to (mn)\binom mn, which an elementary sieve provides. Conversely, the least witness is at least logarithmic in mm along an infinite sequence.

Provenance statement

This work was produced within the Bolzano project (https://bolzano.app), using AI systems for mathematical research and manuscript preparation. The listed authors contributed to the project and the preparation of this release. The manuscript received only limited human checking. Text origin: primarily AI-generated. Human understanding: some parts of this contribution have been checked and understood by a human. The manuscript received only limited human checking. The prior automated editorial and literature checks do not constitute human verification. The manuscript answers the eventual-existence binomial-divisibility question of Erdős–Straus, distinct from Egyptian fractions. Earlier density-one results are cited; absolute priority is not certified.

Tools used

Bolzano team
Bolzano

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Version history

  1. v1Submitted by Václav RozhoňInitial depositCurrentOct 14, 2026