Exact diagonal coefficients of the five-dimensional Geode
Follow the latest progress by email.
Let S be the formal series with constant term 1 satisfying S=1+Σₖ₌₂⁶tₖSᵏ, and define the five-dimensional Geode G by S−1=(t₂+⋯+t₆)G. How can the coefficient of (t₂t₃t₄t₅t₆)^M in G be computed exactly? The concrete challenge is M=1000, where direct summation has about 10¹² terms.
Why it matters and what counts as progress
Why it matters: The five-dimensional challenge tests whether a combinatorial coefficient problem can be reduced below an infeasible direct summation cost.
What counts as progress: Derive an exact recurrence or coefficient-collapse method and compute the target with certified integer reconstruction, rather than floating-point estimation.
How the problem developed
- 13 Aug 2025 · Open problem [1]
Doron Zeilberger and collaborators Posed the five-dimensional Geode challenge asking for an explicit expression for its diagonal coefficients.
- 11 Aug 2026 · Theorem [2]
Jaideep Sai Padhi with Anthropic’s Claude Reported a solution to the five-dimensional challenge and a one-dimensional integral representation for diagonal Geode coefficients. The author says the work was carried out in collaboration with Claude, which generated the mathematical ideas, proofs and verification code.
AI contribution
Solutions to Five Challenge Problems in Enumerative and Algorithmic Combinatorics, with an Account of the Human-Machine Methodology Employed
For the Geode G defined by S−1=(t₂+⋯+t₆)G, a coefficient-collapse identity reduces its diagonal coefficient to a one-dimensional integral and coefficient extraction. It enables an exact 8,367-digit computation of G[1000^(5)] using modular arithmetic and bounded Chinese remainder reconstruction.
Sources
Website timeline
Publication dates and editorial changes.
Claude-assisted method computes the five-dimensional Geode value