Irreducibility of Sharipov’s degree-eight cuboid polynomial

For positive coprime unequal integers a,b, is Sharipov’s degree-eight cuboid polynomial Pₐ,ᵦ(t) irreducible over the integers? This is an irreducibility statement associated with cuboid equations, not a solution of the perfect-cuboid problem.

Why it matters and what counts as progress

Why it matters: A proof resolves this algebraic obstruction within the cuboid framework.

What counts as progress: A proof for every allowed pair, an independently formalized argument, or a nontrivial integer factorization for a valid parameter pair.

How the problem developed

2011 · Conjecture [1]
Rashid Sharipov Introduced the degree-eight polynomial in the first cuboid equations and posed its irreducibility as part of the cuboid conjectures.
13 Oct 2025 · Proof claim [2]
Valery Asiryan Posted an elliptic-curve argument for irreducibility of the two-parameter degree-eight cuboid polynomial over the stated integer range.
20 Aug 2026 · Formal result [3]
Google DeepMind AlphaProof Nexus Released a Lean proof for Sharipov’s first cuboid-polynomial irreducibility conjecture in the positive coprime unequal-parameter case.

AI contribution

AlphaProof Nexus finds a formal cuboid irreducibility proof

For positive coprime unequal integers a and b, an AlphaProof Nexus Lean proof shows that Sharipov’s degree-eight cuboid polynomial Pₐ,ᵦ(t) is irreducible over the integers. Its route uses quartic factor analysis and an infinite descent for x⁴+34x²y²+y⁴=z².

AI involvementAI-autonomous

ImportanceModest

Technical verificationFormal (Lean 4/Mathlib and Comparator + human)

Human understandingFull