Beluhov’s Conjecture on the Tchoukaillon Array

Beluhov’s conjecture describes the quadratic growth of the Tchoukaillon array from its recursion; it was relayed by Knuth and arose from numerical evidence. The new claim proves the asymptotic uniformly in both indices.

Known results

• On the row edge, Erdős–Jabotinsky proved an $O(j^{4/3})$ error bound and conjectured $O(j)$.
• Broline–Loeb later proved the edge estimate $s(n)=n^2/\pi+O(n)$.
• The new preprint says these edge results fit the broader two-dimensional asymptotic.

August 2026 claimed proof

Li and Shisheng claim, uniformly for $i,j\ge 0$ with $N=i+j+1\to\infty$, that $\mathrm{Ч}_{i,j}=\frac{(\pi i+2j+2)^2}{4\pi}+O(N^{4/3})$, implying Beluhov’s conjectured quadratic asymptotic. The paper also derives geometric and algorithmic corollaries, while the sharper $O(N)$ error term remains open. ChatGPT and Claude Code assisted with exposition, experiments, and discussion; they are not credited with independently producing the proof.

Current status (as of August 2026): The conjecture has a new claimed uniform proof with an $O(N^{4/3})$ error term, but remains unverified; the sharper $O(N)$ estimate is open.