A plane trigonometric proof for the case n = 4 of Fermat’s Last Theorem

Giri Prabhakar
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 4, Pages 154–163
DOI: 10.7546/nntdm.2021.27.4.154-163
Full paper (PDF, 221 Kb)

Details

Authors and affiliations

Giri Prabhakar
Siemens Technology
Electronics City, Bangalore 560100, India

Abstract

We present a plane trigonometric proof for the case n = 4 of Fermat’s Last Theorem. We first show that every triplet of positive real numbers (a, b, c) satisfying a⁴ + b⁴ = c⁴ forms the sides of an acute triangle. The subsequent proof is founded upon the observation that the Pythagorean description of every such triangle expressed through the law of cosines must exactly equal the description of the triangle from the Fermat equation. On the basis of a geometric construction motivated by this observation, we derive a class of polynomials, the roots of which are the sides of these triangles. We show that the polynomials for a given triangle cannot all have rational roots. To the best of our knowledge, the approach offers new geometric and algebraic insight into the irrationality of the roots.

Keywords

• Pythagorean theorem
• Diophantine equations
• Fermat’s Last Theorem

2020 Mathematics Subject Classification

• 51N20
• 11D41

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