Finitude of the Fibers of Complementary Bell Numbers

Subbarao and Verma asked in 1999 (Problem 5.7, first part) whether the complementary Bell numbers $f(n) = B_n(-1)$ take any given value only finitely many times. Campbell proves they do: for every fixed integer the fiber is finite, a result whose techniques connect to Wilf's conjecture on the vanishing of $f(n)$ .

Result: Proved
Status: Resolved
AI contribution: AI co-developed
Method: Argument
Field: Enumerative combinatorics
Posed by: M. V. Subbarao, A. Verma
Year posed: 1999
Years open: 27y
Solved: 2026-08-01
Model: GPT-5.6 Pro
Vendor: OpenAI
Collaborators: John M. Campbell
Verification: Unreviewed
Publication: Preprint
Significance: 12 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What the AI did

Solved "through our extensive interactions with GPT-5.6 Pro" during the exploratory and proof-development stages; all AI-generated suggestions were substantially revised, corrected and independently verified by the author.

Source

arXiv

Finitude of the Fibers of Complementary Bell Numbers

What the AI did

Source

Discussion