Gallardo’s irreducibility equivalence for binary all-ones polynomials
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Over F₂, is 1+X+⋯+Xⁿ irreducible exactly when its composition with X²+X+1 is irreducible, for every n≥2? The excluded case n=1 does not obey the same formulation.
Why it matters and what counts as progress
Why it matters: The equivalence connects a sparse algebraic substitution with irreducibility of a familiar binary polynomial family.
What counts as progress: Prove the equivalence using a composition criterion, formalize both directions or classify analogous substitutions.
How the problem developed
- 23 Dec 2019 · Conjecture [1]
Luis H. Gallardo Conjectured that for n≥2, the binary all-ones polynomial is irreducible exactly when the corresponding polynomial in X²+X+1 is irreducible.
GPT-6 Astra, Claude Fable 5.1, and Claude Opus agents, with Brandon Yates selecting and reviewing the target A Lean-verified record gives the per-n equivalence for n≥2 and the corresponding sequence statement, excluding n=1; the proof uses a classical trace criterion.
AI contribution
AI proof of Gallardo’s all-ones polynomial equivalence
For every n≥2 over GF(2), the polynomial 1+X+⋯+X^n is irreducible exactly when its composition with X²+X+1 is irreducible. The formalization also checks the corresponding OEIS set statement and excludes n=1.
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