MathDB

Factorial asymptotics of the Matryoshka numbers

About 12 years old · traced to

Let (an)n≥1(a_n)_{n\ge 1} be the Matryoshka numbers of Definition 1.1 in Liu's note. Then there exists a real number SS with 0<S<∞0<S<\infty such that

lim⁡n→∞an(n+4)!=S,\lim_{n\to\infty}\frac{a_n}{(n+4)!}=S, lim⁡n→∞ann!n4=S,\lim_{n\to\infty}\frac{a_n}{n!n^4}=S,

and

lim⁡n→∞anSn!n4=1.\lim_{n\to\infty}\frac{a_n}{S n!n^4}=1.

Moreover,

0.00542831750≤S≤0.00542831848.0.00542831750\le S\le 0.00542831848.
References

Primary source

Open Math Problems Claimed to Be Solved with AI

mathematical-physics
Posted

Progress summary

Refreshed
Claimed solved

An August 2026 paper reports a complete proof of the conjectured growth law and a highly precise value for its constant, but the proof has not been independently verified.

The problem asks whether the recursively defined Matryoshka numbers have factorial growth with a fourth-degree polynomial factor and a positive limiting constant. Kotěšovec conjectured this from numerical data; it was later restated as Conjecture 5.7 in the cosmohedron literature.

August 11, 2026 claimed proof

The paper Danus: Orchestrating Mathematical Reasoning Agents with Fact-Graph Memory reports that Danus produced a complete proof, establishing the stated limits and the enclosure 0.00542831750≤S≤0.005428318480.00542831750\le S\le 0.00542831848. It says Jihao Liu’s 2026 note contains the corresponding proof. The result is claimed rather than independently verified; the paper attributes the work to the Danus system, which used named models including GPT-5.5 and Claude Opus 4.8.

Current status (as of September 2026): The asymptotic formula and numerical enclosure are claimed proved by Liu and the Danus paper, but remain unverified pending independent mathematical corroboration.

Sources

Solutions 0

No solutions have been posted yet.