The small Davenport constant of the Heisenberg group of order 125

Patrick White

For the exponent-five Heisenberg group of order 125, the small unordered Davenport constant is twelve: x⁴y⁴v⁴ is product-one-free, and every thirteen-term sequence has a product-one subsequence in some ordering.

Sources for this record

What was already known

Godara and Sarkar proved the p=3 value six and conjectured 3p−3 for odd primes. The p=5 case was open; the ordered invariant has a different value.

AI involvement AI-led

White assigns the principal research role to AI systems under his direction. Claude developed framing and verification code; GPT-5.5 supplied the corrected reduction.

The attribution note explicitly assigns the principal research role and the retained upper-bound reduction to AI systems under human direction; it does not establish AI-autonomous operation.

Importance Notable

Would settle the first open case of the Heisenberg small Davenport-constant conjecture.

Limitations

The exact finite searches check the p=5 spread condition, while no completed check of the full reduction is documented. The result does not cover p≥7 or the ordered constant.

Technical verification Partial

The released repository provides C and Python exhaustive spread checks and auxiliary finite checks. No complete check of the handwritten reduction is documented, so the universal claim remains Partial.

Human understanding Few

The mathematical explanation and write-up are credited to Claude; no named human explains the central spread-bound mechanism. Few is the evidence default.

Correction and publication history

2026-10-18 · publication

Would settle the first open case of the Heisenberg small Davenport-constant conjecture.

Read this published revision