Two-Variable Factorial Conjecture

Let $\mathcal{L}(x^{a}y^{b})=a!\,b!$ on $\mathbb{C}[x,y]$. The Factorial Conjecture asks whether $\mathcal{L}(f^{m})=0$ for every $m\geq 1$ forces $f=0$. The homogeneous two-variable case was settled by Liu and Sun; the inhomogeneous problem does not reduce to it, because radial integration couples the homogeneous layers through Gamma factors. A claimed proof settles the full two-variable case affirmatively.

Result

Proved (Claimed in a self-published research draft; a standalone by-product is the transcendence of the integral of exp(q) between distinct algebraic endpoints for nonconstant algebraic q)

Status

Candidate (review pending)

AI contribution

AI co-developed

Method

Argument

Field

Commutative Algebra, Transcendence

Posed by

Arno van den Essen, David Wright, Wenhua Zhao

Year posed

2011

Years open

15y

Solved

2026-08-01

Model

GPT-5.6 Sol, Claude Opus 5

Vendor

OpenAI, Anthropic

Collaborators

Christopher D. Long

Verification

Unreviewed

Publication

Announced

Significance

20 / 100*

Disclosed cost

—

Wikipedia

No dedicated article