A Resolution of Erdős Problem 731 under Dyadic Regularity

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 $X\le n&lt;2X$, put $L=\log(2X)$ and $${\mathcal F}<em>X=\sqrt2(\log2)<sup>{1/4}L<sup>{1/4}\exp\sqrt{(\log2)L}$$. Uniformly for $1\le z\le Z(X)=o(L<sup>{1/4})$, we prove $${\mathbb P}_X(A(n)\le {\mathcal F}_X\exp(-z))\asymp \exp(-2z)$$ and ${\mathbb P}_X(A(n)&gt;{\mathcal F}_X\exp(z))\ll \exp(-2z)$. Consequently $$\log A(n)=\sqrt{(\log2)\log n}+\frac14\log\log n+O</em>{\rm dens}(1)$$. 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.

• The paper establishes the precise second-order asymptotic scale for A(n) using a novel moving-base large sieve method.
• It demonstrates that no dyadically regular deterministic function can serve as an asymptotic equivalent for A(n) in natural density.
• The advanced variance analysis and mesoscopic tail bounds reveal the inherent fluctuations in the divisibility of central binomial coefficients.

Resolution of Erdős Problem 731 under Dyadic Regularity

Background and Problem Statement

The paper "A Resolution of Erdős Problem 731 under Dyadic Regularity" (2606.29062) investigates the behavior of the least positive integer $A(n)$ not dividing the central binomial coefficient $B_n = \binom{2n}{n}$:

$A(n) = \min\{ m \ge 1 : m \nmid \binom{2n}{n} \}.$

Erdős Problem 731 asks for a "reasonable" deterministic function $f(n)$ such that $A(n)\sim f(n)$ for almost all $n$ in the sense of natural density, meaning $A(n)/f(n) \to 1$ in natural density. This problem had previously only been resolved at a first-order level, with prior heuristics suggesting that $\log A(n)$ typically lies in a window of width $o(\sqrt{\log n})$ about $\sqrt{(\log 2)\log n}$, but with neither leading constants nor tightness results.

The result here rigorously establishes both the precise second-order scale for $B_n = \binom{2n}{n}$0 and demonstrates that no dyadically regular deterministic function can serve as an asymptotic equivalent for $B_n = \binom{2n}{n}$1 in natural density. The analysis leverages additive combinatorial large sieve bounds, Kummer's theorem, and a refined variance analysis for the digit structure in varying prime bases.

Main Results and Structural Overview

Density-Tight Logarithmic Scale

The key formula for the density-tight scale is

$B_n = \binom{2n}{n}$2

The main distributional result is that for any function $B_n = \binom{2n}{n}$3,

$B_n = \binom{2n}{n}$4

for almost all $B_n = \binom{2n}{n}$5; equivalently,

$B_n = \binom{2n}{n}$6

Here $B_n = \binom{2n}{n}$7 denotes tightness in natural density, i.e., for every diverging window, the deviation remains within the window for almost all $B_n = \binom{2n}{n}$8.

Dyadic Regularity and the Negative Result

Dyadic regularity is defined as follows: a function $B_n = \binom{2n}{n}$9 is dyadically regular if, for every dyadic interval $A(n) = \min\{ m \ge 1 : m \nmid \binom{2n}{n} \}.$0, the variation of $A(n) = \min\{ m \ge 1 : m \nmid \binom{2n}{n} \}.$1 over that block tends to zero with $A(n) = \min\{ m \ge 1 : m \nmid \binom{2n}{n} \}.$2. This class absorbs all standard "smooth" or "slowly varying" functions constructed from powers and iterates of logarithms.

The nonconcentration theorem states that even after optimal centering, $A(n) = \min\{ m \ge 1 : m \nmid \binom{2n}{n} \}.$3 exhibits persistent multiplicative nonconcentration on every sufficiently large dyadic block. Specifically, there exist constants $A(n) = \min\{ m \ge 1 : m \nmid \binom{2n}{n} \}.$4 and $A(n) = \min\{ m \ge 1 : m \nmid \binom{2n}{n} \}.$5 such that for all large $A(n) = \min\{ m \ge 1 : m \nmid \binom{2n}{n} \}.$6,

$A(n) = \min\{ m \ge 1 : m \nmid \binom{2n}{n} \}.$7

where $A(n) = \min\{ m \ge 1 : m \nmid \binom{2n}{n} \}.$8 equals $A(n) = \min\{ m \ge 1 : m \nmid \binom{2n}{n} \}.$9 up to an $f(n)$0 scaling in the exponent. Consequently, no dyadically regular function $f(n)$1 can satisfy $f(n)$2 in density.

Quantitative Mesoscopic Tail Bounds

Let $f(n)$3, and define $f(n)$4, $f(n)$5, and for $f(n)$6,

$f(n)$7

This gives two-sided control for the lower tail and one-sided for the upper, and fully describes the mesoscopic fluctuations around the centering scale.

Methodological Innovations

The analysis builds upon several classical and modern techniques, including:

• Kummer's Theorem: $f(n)$8 equals the number of carries when adding $f(n)$9 in base $A(n)\sim f(n)$0. For odd $A(n)\sim f(n)$1, $A(n)\sim f(n)$2 iff all base-$A(n)\sim f(n)$3 digits of $A(n)\sim f(n)$4 are less than $A(n)\sim f(n)$5.
• Carry-Deficit Reformulation: Expressing the event $A(n)\sim f(n)$6 in terms of the least common multiple, $A(n)\sim f(n)$7 dividing $A(n)\sim f(n)$8.
• Moving-Base, Growing-Depth Variance: The main technical novelty is a strip variance estimate, uniform in both base and depth, executed via the additive large sieve in the moving-base setting. Smooth approximations to digit restrictions yield tractable variance estimates, crucial for anti-concentration on mesoscopic scales.
• Mesoscopic Paley–Zygmund Application: The lower tail probability bounds are obtained via precise mean and variance analysis for the sum of indicator variables over primes in certain intervals (strips).

Context within Prior Work

Previous results, including those by Erdős, Graham, Ruzsa, and Straus [EGRS75], only established, for every fixed $A(n)\sim f(n)$9, $n$0 for almost all $n$1, without tight asymptotics or constants. Further discussions, e.g., by Pomerance [Pomerance15, Pomerance26], as well as the online heuristic formulated by Zeraoulia Rafik in April 2026, anticipated certain features (e.g., Poisson-like behavior under independent base assumptions), but could not control the correlation phenomena essential for nonconcentration proofs.

Implications and Theoretical Significance

Failure of Concentration: No Asymptotic Equivalent

The negative result regarding dyadic regularity is particularly strong: the least nondivisor $n$2 cannot be captured by any reasonable deterministic function in density, due to essential fluctuations induced by the underlying digit structure in varying prime bases. The randomness in the least nondivisor is an unavoidable, not an artifact of centering. Any possible "equivalent" function $n$3 must itself inherit the same type of blockwise oscillation, which rules out traditional slowly varying or smooth scaling.

Precise Asymptotic Scale in Density

The determination of the explicit second-order term for $n$4 and the matching tail probabilities gives a sharp description of typical behavior. This level of quantitative control should have broader application to similar problems involving random digit restrictions or carries in non-fixed bases.

Methodological Impact

The moving-base large sieve variance theorem introduces an approach to digit-restriction and carry-based phenomena not previously exploited in this setting. The smooth minorants and precise control over both mean and variance across variable bases and growing digit depth open new directions in the analysis of arithmetical functions with "digit randomness"—with potential relevance in random matrix theory, computational complexity, and analysis of cryptographic primitives.

Pathways for Further Research

The sequels to this work are clearly demarcated: the results stop short of proving a limiting law (e.g., Poisson or other universality class), or supplying a limiting normalization for the distribution of $n$5. The nonconcentration result, however, places strong constraints on any such limiting distributions, and further analysis may bridge the gap to a full distributional characterization. Additionally, the variance techniques developed could be generalized to other divisor and arithmetic obstruction problems for combinatorially structured sequences.

Conclusion

The paper provides a complete density-asymptotic answer to Erdős Problem 731 under dyadic regularity. The scale $n$6 represents the optimal deterministic normalization within the density-tight regime, and the result settles, in the negative, the possibility of asymptotic equivalence to any dyadically regular function. The technical device of moving-base variance estimation via the additive large sieve is of independent methodological interest. This resolves a long-standing open problem and establishes that the least nondivisor of $n$7 exhibits unavoidable mesoscopic randomness on any sufficiently large dyadic interval. Future work may address the finer limiting distributional questions now highlighted by this analysis.