Solutions to Five Challenge Problems in Enumerative and Algorithmic Combinatorics, with an Account of the Human-Machine Methodology Employed

Jaideep Sai Padhi

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 for this record

What was already known

Amdeberhan, Kauers and Zeilberger posed the D=5, M=1000 computation. Rubine solved D=4 and estimated that the direct five-dimensional sum, with about 10¹² terms, needed a new method.

AI involvement AI-led

Claude supplied the collapse, algorithm and code. Padhi chose the challenge, ran the calculation, designed checks and corrected model errors.

Claude supplied the collapse, algorithm and code; Padhi chose the challenge, sized and executed the computations and applied independent controls.

Importance Notable

Reports a method and exact value for the specified five-dimensional Geode challenge.

Limitations

Fresh-prime and small-case controls support the finite computation but do not independently prove the general coefficient-collapse identity. The separate four-variable recurrence challenge remains open.

Technical verification Partial

The finite coefficient is reported as exact via modular polynomial evaluation, a proved magnitude bound and CRT reconstruction; fresh-prime checks plus D=4 and small-value controls support the computation. These checks do not independently establish the preceding coefficient-collapse identity, so the complete assessed method is only partially checked.

Independent checking

Exact modular quadrature and coefficient recurrence, bounded CRT reconstruction, fresh-prime checks and published small-case controls

Checker
Jaideep Sai Padhi, paper author, executing model-written code
Scope
G[1000^(5)] exact finite coefficient
Basis
Reported by the linked checker
Evidence source

Human understanding Few

The author designed and ran the checking protocol, but the source does not document his independent explanation of Claude’s coefficient-collapse mechanism; Few is the evidence default.

Correction and publication history