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arxiv: 2606.29062 · v1 · pith:ZEY3AXHRnew · submitted 2026-06-27 · 🧮 math.NT

A Resolution of ErdH{o}s Problem 731 under Dyadic Regularity

Eric Li (Trinity College , University of Cambridge) This is my paper

Pith reviewed 2026-06-30 08:03 UTC · model grok-4.3

classification 🧮 math.NT
keywords Erdős problem 731A(n) binomial non-divisordyadic regularitynatural densitylogarithmic asymptoticvariance theoremcentral binomial coefficient
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The pith

Under dyadic regularity, log A(n) equals sqrt((log 2) log n) plus one-fourth log log n plus a density-bounded term.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper resolves Erdős Problem 731 by determining the size of A(n), the smallest positive integer not dividing the central binomial coefficient binom(2n,n), under an explicit formalization of dyadic regularity. It proves that on dyadic intervals the distribution of A(n) follows specific tail probabilities around an explicit scale F_X, which directly yields the logarithmic asymptotic formula. A reader would care because this provides the first conditional resolution of the problem and demonstrates that A(n) cannot be captured by any single deterministic function in natural density. The approach keeps the exact divisibility condition from the least common multiple while introducing a restricted-digit variance theorem to replace heuristic independence.

Core claim

We resolve Erdős 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≤n<2X, put L=log(2X) and F_X=sqrt(2)(log2)^{1/4}L^{1/4}exp sqrt((log2)L). Uniformly for 1≤z≤Z(X)=o(L^{1/4}), we prove P_X(A(n)≤F_X exp(-z))≍exp(-2z) and P_X(A(n)>F_X exp(z))≪exp(-2z). Consequently log A(n)=sqrt((log2)log n)+(1/4)log log n + O_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)→1 in natural density.

What carries the argument

The moving-base restricted-digit variance theorem that establishes the tail estimates while retaining the exact least-common-multiple divisibility condition for membership in the set of divisors of binom(2n,n).

If this is right

  • The logarithmic asymptotic for A(n) holds with an error term that is bounded in natural density.
  • The probability that A(n) is much smaller or larger than the central scale F_X decays exponentially as exp(-2z).
  • No dyadically regular deterministic function f can satisfy A(n) ~ f(n) in natural density.
  • The result applies uniformly across all sufficiently large dyadic intervals without additional assumptions on the sequence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the regularity condition holds for typical sequences, then A(n) must fluctuate multiplicatively on logarithmic scales.
  • Similar variance theorems could apply to other problems involving the prime factors of central binomial coefficients.
  • The nonconcentration implies that the set of n where A(n) is close to any given scale has density zero.

Load-bearing premise

The dyadic-regularity condition ensures that the tail probabilities for A(n) around F_X match the stated exponential bounds uniformly for z up to o(L^{1/4}).

What would settle it

An explicit computation over a large dyadic interval [X,2X] where the measured proportion of n with A(n) > F_X exp(z) for fixed z significantly exceeds the upper bound exp(-2z) times a constant.

read the original abstract

We resolve Erdos 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 ${\mathcal F}_X=\sqrt2(\log2)^{1/4}L^{1/4}\exp\sqrt{(\log2)L}$. 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript resolves Erdős Problem 731 under an explicit dyadic-regularity formalization of 'reasonable.' Let A(n) be the least positive integer not dividing binom(2n,n). On dyadic intervals X ≤ n < 2X, with L = log(2X) and F_X = sqrt(2)(log 2)^{1/4} L^{1/4} exp(sqrt((log 2)L)), it proves the uniform tail bounds P_X(A(n) ≤ F_X exp(-z)) ≍ exp(-2z) and P_X(A(n) > F_X exp(z)) ≪ exp(-2z) for 1 ≤ z ≤ Z(X) = o(L^{1/4}). These yield the asymptotic log A(n) = sqrt((log 2) log n) + (1/4) log log n + O_dens(1) and the dyadic non-concentration result that no dyadically regular deterministic scale f satisfies A(n)/f(n) → 1 in natural density. The proof retains the exact LCM divisibility condition on binomial coefficients and replaces cross-base independence heuristics by a moving-base restricted-digit variance theorem.

Significance. If the variance theorem and tail bounds hold, the work supplies a conditional but precise resolution of a longstanding problem in combinatorial number theory, with an explicit centering function F_X, uniform tails up to z = o(L^{1/4}), and a strong negative result ruling out dyadically regular deterministic approximations. The methodological replacement of heuristics by an explicit variance theorem and the retention of the exact divisibility condition are notable strengths; the O_dens(1) error and the non-concentration statement are direct consequences of the stated probability bounds.

minor comments (3)
  1. The notation O_dens(1) is used in the abstract and conclusion but should be defined explicitly (e.g., as an error term that holds outside a set of density zero) at its first appearance.
  2. The formal definition of 'dyadically regular' is invoked throughout but would benefit from a self-contained statement in the introduction or §1 before the statement of the main theorems.
  3. In the derivation of the asymptotic from the tail bounds, the transition from dyadic intervals to the density statement requires an explicit summation or averaging argument over the dyadic scales; this step should be expanded for clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of the methodological contributions, and recommendation for minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation chain begins from the exact LCM divisibility condition on binomial coefficients together with a new moving-base restricted-digit variance theorem that replaces prior heuristics. The uniform tail bounds P_X(A(n) ≤ F_X exp(-z)) ≍ exp(-2z) and the complementary upper tail are stated as proved results on dyadic intervals; the centering F_X and the consequent asymptotic log A(n) = sqrt((log2) log n) + (1/4) log log n + O_dens(1) are direct analytic consequences of those bounds rather than fitted inputs or self-definitions. Dyadic non-concentration is likewise derived from the same tail statements. No load-bearing self-citations, ansatzes smuggled via citation, uniqueness theorems imported from the authors' prior work, or renaming of known empirical patterns appear in the provided text. The restriction to dyadic regularity is declared explicitly as the formalization under which the statements hold.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard properties of binomial coefficients and least common multiples; no free parameters, ad-hoc axioms, or invented entities are introduced. The new variance theorem is presented as an external tool rather than postulated.

axioms (1)
  • standard math Standard properties of binomial coefficients and least common multiples for the divisibility condition
    The proof retains the exact least-common-multiple divisibility condition.

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