Elementary Polynomial-Roots Approach to Fermat's Last Theorem — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Fermat's Last Theorem (FLT) — no integer solutions for \(x^n + y^n = z^n\), \(n>2\) — proven via elliptic curves and modular forms; a polynomial-roots approach is proposed as an elementary alternative. MATH: Core equation: \(a^n + b^n = c^n\) (no nonzero integers for \(n>2\)). Wiles' proof: every semistable elliptic curve \(E: y^2 = x^3 + Ax + B\) is modular (Taniyama–Shimura for semistable case). Frey curve: \(y^2 = x(x - a^p)(x + b^p)\) — if FLT counterexample existed, this curve would be non-modular, contradiction. Polynomial approach (arXiv:1105.0669): associate \(P(x) = x^n - (a^n + b^n)\) or similar; roots studied to show no integer \(c\) exists. Key constants: none new — but the modular discriminant \(\Delta = -16(4A^3 + 27B^2)\) and \(j\)-invariant appear. CONNECTION: The Frey curve's discriminant is \(2^{-8}(abc)^{2p}\) — a perfect power, linking to lattice/root-system structure (the \(E_8\) lattice and Weyl group appear in modular forms). The proof's core is a sy Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204485433
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