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 *a*^{4} + *b*^{4} = *c*^{4} 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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## Cite this paper

Prabhakar, G. (2021). A plane trigonometric proof for the case *n* = 4 of Fermat’s Last Theorem. *Notes on Number Theory and Discrete Mathematics*, 27(4), 154-163, DOI: 10.7546/nntdm.2021.27.4.154-163.