Erdős Problem #731 — Find some reasonable function $f(n)$ such that, for almost all integers $n$, the least integer $m$ such that $m mid \binom{2n}{n}$ satisfies $m\sim f(n).$

Erdős, Graham, Ruzsa, and Straus posed the problem in 1975: determine whether the least integer failing to divide the central binomial coefficient is asymptotic to a reasonable function for almost all inputs.

Known results

• Erdős, Graham, Ruzsa, and Straus (1975) recorded the coarse estimate $A(n)=\exp\!\big((\log n)^{1/2+o(1)}\big)$ for almost all $n$.

June 2026 dyadic-regularity resolution

A preprint proves that no dyadically regular function $f$ can satisfy $A(n)/f(n)\to1$ in natural density. It also establishes the density-tight scale $F(n)=\sqrt{2}(\log 2)^{1/4}(\log n)^{1/4}\exp\!\big(\sqrt{(\log 2)\log n}\big)$, while proving persistent multiplicative spread on every sufficiently large dyadic block. The work is formally verified in Lean $4$, but its negative conclusion applies only to this explicit formalization of “reasonable.”

Current status (as of June 2026): The problem is resolved negatively for dyadically regular $f$, but the original question without that restriction remains open.