A Resolution of Erd\H{o}s Problem 731 under Dyadic Regularity

Abstract

We resolve Erd\H{o}s Problem 731 under the explicit dyadic-regularity formalization of "reasonable." Let $A(n)$ be the least positive integer not dividing $\binom{2n}{n}$. On dyadic intervals $X\le n<2X$, put $L=\log(2X)$ and $F_X=\sqrt{2}(\log 2)^{1/4}L^{1/4}\exp\sqrt{(\log 2)L}$. Uniformly for $1\le z\le Z(X)=o(L^{1/4})$, we prove $\mathbb{P}_X(A(n)\le F_X\exp(-z))\asymp \exp(-2z)$ and $\mathbb{P}_X(A(n)>F_X\exp(z))\ll \exp(-2z)$. Consequently $\log A(n)=\sqrt{(\log 2)\log n}+\frac{1}{4}\log\log n+O_{\mathrm{dens}}(1)$. We also prove dyadic nonconcentration: no scalar center on a large dyadic block, and hence no dyadically regular deterministic scale $f$, can satisfy $A(n)/f(n)\to 1$ in natural density. The proof retains the exact least-common-multiple divisibility condition and replaces heuristic cross-base independence by a moving-base restricted-digit variance theorem. The resolution proved here has been formally verified in Lean.