The purpose of the following report is to demonstrate that with the mathematical tools of the 17th century, a proof to Fermat’s conjecture may have been possible. The whole procedure is based on planar geometry, which was perfectly known at the time. Moreover, the procedure does not exceed contemporary school mathematics.
Fermat´s last theorem has been a hot topic for mathematicians since it was originally published in Fermat’s "Arithmetica". Although Pierre de Fermat had claimed to have found an easy proof for his conjecture, he failed to disclose it. Over the past 350 years, mathematicians were unable to find this simple proof. So, over time, it was concluded that Fermat had made an error with his claim. Finally, in 1994 the British mathematician Andrew Wiles found a complex mathematical proof for Fermat’s conjecture.
Table of Contents
1 Introduction
2 Rules for Even, Odd and Rational Numbers
3 Fermat’s Equation
4 Fermat’s Triangle
5 Hero’s Triangle
6 What are the Angles?
7 Conclusion
8 Triangle Area
9 Summary
10 Bibliography
Objectives and Topics
This report aims to demonstrate that a proof for Fermat’s Last Theorem might have been achievable using the planar geometry tools available in the 17th century, potentially offering a simpler alternative to modern complex proofs.
- Mathematical foundation using set theory and arithmetic rules for even, odd, and rational numbers.
- Transformation of Fermat’s equation into a geometric representation using the "Fermat triangle".
- Application of trigonometric laws and Thales’ theorem to analyze the properties of this triangle.
- Logical deduction showing the impossibility of the theorem for exponents greater than 2.
Excerpt from the Book
3 Fermat’s Equation
Fermat’s conjecture claims that the n-th power of a natural number cannot be expressed as the sum of two other integers with the same power if the exponent is greater than 2. This means that if n > 2 there is no solution for a^n + b^n = c^n (3.1) with (a, b, c) ∈ ℕ. Note that c is always the largest number in triple (a, b, c).
The case of n = 1 is elementary because any integer may be expressed as sum of two other (smaller) integers. Therefore, we can restrict our considerations to cases when n ≥ 2 (3.2).
Assuming that (3.1) is valid for any n we shall simplify equation (3.1) by removing all common multipliers without forfeiting generality. There shall be no common divisor remaining in triple (a, b, c).
Combination a = b is no valid solution of (3.1) because a and b have a common divisor “a”. Therefore, without loss of generality we define a < b < c (3.3).
Summary of Chapters
1 Introduction: Provides an overview of the challenge and establishes the mathematical notation and number sets used in the report.
2 Rules for Even, Odd and Rational Numbers: Defines the specific calculation rules for addition and multiplication of different number types necessary for the geometric proof.
3 Fermat’s Equation: Formulates the mathematical constraints for Fermat’s conjecture and defines the necessary conditions for the investigated integer triples.
4 Fermat’s Triangle: Introduces the concept of representing Fermat’s equation as a planar triangle and applies the law of sines and cosines.
5 Hero’s Triangle: Utilizes Hero’s formula for the area of a triangle to verify if the area of the defined Fermat triangle is rational.
6 What are the Angles?: Analyzes the angles of the triangle to establish a connection with Thales’ theorem.
7 Conclusion: Successfully derives a contradiction to Fermat’s original equation, proposing the proof for higher exponents.
8 Triangle Area: Refines the proof by examining the properties of the triangle area and its parity.
9 Summary: Recaps the core steps of the geometrical proof and asserts the feasibility of this school-level approach.
10 Bibliography: Lists the referenced mathematical concepts and historical definitions used throughout the study.
Keywords
Fermat’s Last Theorem, Pierre de Fermat, Andrew Wiles, Planar Geometry, Fermat Triangle, Thales’ Theorem, Hero’s Formula, Even Numbers, Odd Numbers, Rational Numbers, Trigonometric Functions, Integer Triples, Mathematical Proof.
Frequently Asked Questions
What is the core subject of this paper?
The paper investigates a potential 17th-century mathematical approach to proving Fermat’s Last Theorem using elementary planar geometry.
What are the central themes discussed?
The central themes include number properties (even/odd), geometric representation of equations, trigonometric laws, and the use of Thales’ theorem in proofs.
What is the primary goal of the author?
The goal is to demonstrate that a simple proof for Fermat’s conjecture, which has historically been considered elusive, may be attainable using geometry known during Fermat’s lifetime.
Which scientific methodology is employed?
The author uses deductive logic based on classical planar geometry and elementary school-level algebraic manipulations.
What is covered in the main section?
The main section establishes the rules for numbers, defines a "Fermat triangle", and utilizes trigonometric identities to identify contradictions in the conjecture.
How is the work characterized by typical keywords?
It is characterized by terms such as Fermat's Last Theorem, planar geometry, integer triples, and trigonometric identities.
What role does the "Fermat triangle" play in the proof?
It allows the author to translate the algebraic equation a^n + b^n = c^n into a physical shape where laws of sines and cosines can be applied.
How does the author use Thales' theorem?
The author demonstrates that solutions to Fermat’s equation must exist on a Thales’ semi-circle, which simplifies the expression to a known Pythagorean relationship.
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- Hans Georg Schrey (Autor:in), 2023, A 17th Century Approach to Fermat's Last Theorem, München, GRIN Verlag, https://www.grin.com/document/1396819
