Eventual membership in the Erdős–Straus divisibility set

Determine whether all sufficiently large integers m belong to G_n; equivalently, for every fixed integer n at least 2, determine whether every sufficiently large m admits an integer k with 1≤k≤m−n such that the binomial coefficient (n+k choose n) divides (m+k choose k).

Background

The paper proves that G_n has natural density one for every fixed n≥2, but density one does not imply that its complement is finite. The authors therefore leave unresolved the stronger assertion that every sufficiently large m satisfies the required divisibility condition.

References

Whether all large m lie in Gn remains open, since we show that N \ Gn has density zero and not that it is finite.

— The Problem Is the Problem: Towards Scalable Mathematical Discovery  (2608.16977 - Zheng et al., 17 Aug 2026) in Remark C.5.2, Appendix C.5

That case is raised in Erd˝os & Straus (1977), recorded by Erd˝os & Graham (1980), and appears as Problem 389 on the Erd˝os problems website (Bloom, 2026), where it is listed as open.

— The Problem Is the Problem: Towards Scalable Mathematical Discovery  (2608.16977 - Zheng et al., 17 Aug 2026) in Section C.5.1 and Remark C.5.2, Appendix C.5