Fermat's Last Theorem: An Elementary Proof via Factorization and the Reconstruction of Fermat's Original Insight
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2025
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| author | ACOSTA PADILLA, ALFREDO LUIS |
| author_facet | ACOSTA PADILLA, ALFREDO LUIS |
| contents | <p>We present a complete elementary proof of Fermat's Last Theorem using only basic algebra and the classical factorization of odd powers. By normalizing Fermat's equation through division by z, we obtain irreducible fractions that must sum to an integer. For odd exponents n≥3, the natural factorization x^n + y^n = (x+y)·Q(x,y) creates an insurmountable magnitude obstruction that prevents integer solutions. The proof requires only undergraduate mathematics—factorization, coprimality, and magnitude analysis—making it accessible to any mathematics student.</p> <p>More significantly, we reconstruct the historical context of Fermat's famous marginal note. We argue that his "truly marvelous proof" referred specifically to the case n=4 (proven via infinite descent, his only documented proof), while odd exponents were geometrically trivial for a 17th-century mathematician and required no documentation. The 358-year search for a "lost proof" was based on a misinterpretation of a personal reminder as a public announcement.</p> <p>This work demonstrates that Fermat's Last Theorem does not require the advanced machinery of Wiles' 150-page proof (elliptic curves, modular forms, Galois representations), but follows directly from elementary factorization principles known in Fermat's time. The tragedy was not that Fermat's proof was lost—it was that algebraic intuition was lost.</p> <p>Key results:<br>- Complete elementary proof for all n≥3 <br>- Factorization argument for odd exponents (the missing piece)<br>- Historical reconstruction of Fermat's actual reasoning<br>- Demonstration that Wiles' approach, while mathematically profound, was unnecessary for FLT specifically</p> <p>This paper makes Fermat's Last Theorem accessible to undergraduate students and vindicates Fermat's claim that he possessed an elementary proof.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17687924 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
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| spellingShingle | Fermat's Last Theorem: An Elementary Proof via Factorization and the Reconstruction of Fermat's Original Insight ACOSTA PADILLA, ALFREDO LUIS Fermat's Last Theorem Elementary proof Diophantine equations Factorization Number theory Mathematical history Pierre de Fermat Andrew Wiles Algebraic number theory Coprime integers Primitive solutions Magnitude obstruction Cyclotomic polynomials 17th century mathematics Alternative proofs <p>We present a complete elementary proof of Fermat's Last Theorem using only basic algebra and the classical factorization of odd powers. By normalizing Fermat's equation through division by z, we obtain irreducible fractions that must sum to an integer. For odd exponents n≥3, the natural factorization x^n + y^n = (x+y)·Q(x,y) creates an insurmountable magnitude obstruction that prevents integer solutions. The proof requires only undergraduate mathematics—factorization, coprimality, and magnitude analysis—making it accessible to any mathematics student.</p> <p>More significantly, we reconstruct the historical context of Fermat's famous marginal note. We argue that his "truly marvelous proof" referred specifically to the case n=4 (proven via infinite descent, his only documented proof), while odd exponents were geometrically trivial for a 17th-century mathematician and required no documentation. The 358-year search for a "lost proof" was based on a misinterpretation of a personal reminder as a public announcement.</p> <p>This work demonstrates that Fermat's Last Theorem does not require the advanced machinery of Wiles' 150-page proof (elliptic curves, modular forms, Galois representations), but follows directly from elementary factorization principles known in Fermat's time. The tragedy was not that Fermat's proof was lost—it was that algebraic intuition was lost.</p> <p>Key results:<br>- Complete elementary proof for all n≥3 <br>- Factorization argument for odd exponents (the missing piece)<br>- Historical reconstruction of Fermat's actual reasoning<br>- Demonstration that Wiles' approach, while mathematically profound, was unnecessary for FLT specifically</p> <p>This paper makes Fermat's Last Theorem accessible to undergraduate students and vindicates Fermat's claim that he possessed an elementary proof.</p> |
| title | Fermat's Last Theorem: An Elementary Proof via Factorization and the Reconstruction of Fermat's Original Insight |
| topic | Fermat's Last Theorem Elementary proof Diophantine equations Factorization Number theory Mathematical history Pierre de Fermat Andrew Wiles Algebraic number theory Coprime integers Primitive solutions Magnitude obstruction Cyclotomic polynomials 17th century mathematics Alternative proofs |
| url | https://doi.org/10.5281/zenodo.17687924 |