A Resolution of Erdős Problem 731 under Dyadic Regularity

Abstract

We resolve Erdos Problem 731 under the explicit dyadic-regularity formalization of &#34;reasonable.&#34; 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 ${\mathcal F}_X=\sqrt2(\log2)^{1/4}L^{1/4}\exp\sqrt{(\log2)L}$. Uniformly for $1\le z\le Z(X)=o(L^{1/4})$, we prove ${\mathbb P}_X(A(n)\le {\mathcal F}_X\exp(-z))\asymp \exp(-2z)$ and ${\mathbb P}_X(A(n)>{\mathcal F}_X\exp(z))\ll \exp(-2z)$. Consequently $\log A(n)=\sqrt{(\log2)\log n}+\frac14\log\log n+O_{\rm dens}(1)$. We also prove dyadic nonconcentration: no scalar center on a large dyadic block, and hence no dyadically regular deterministic scale $f$, can satisfy $A(n)/f(n)\to1$ 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.