04 · Mathematics · Research
Personal research into an open irreducibility problem involving a deceptively simple family of Fibonacci polynomials.
Context
Mathematics is still something I actively practise. The Garth–Mills–Mitchell conjecture concerns the polynomial family below, where F denotes the Fibonacci numbers. It states that every polynomial in the family is irreducible over the rational numbers: it cannot be decomposed into simpler polynomial factors with rational coefficients. The definition is elementary, but proving the claim in every degree appears surprisingly difficult.
How I approach it
I combine proof-based reasoning with computation and AI-assisted exploration. AI acts as a research companion: it helps me examine possible directions, challenge arguments, expose gaps, design experiments, compare reformulations, organise a branching investigation, and turn numerical patterns into candidate lemmas or better questions. It can suggest structure and test the resilience of an idea; mathematical validity still depends on rigorous proof and independent verification.
Why it matters
I enjoy the contrast between a problem that can be stated in a few lines and the depth that appears when one tries to prove it. It is also an unusually good setting for exploring what language models can and cannot contribute to open-ended mathematical reasoning—not just productivity or code, but a genuinely difficult intellectual process.
Current status
Active personal research. The investigation is ongoing and unpublished; no resolution of the conjecture is claimed.