We show here, by a novel process, that the infinite sets of integer triples which are individually unique and which derive from Pythagoras’ equation together with those from the linear Diophantine equation, x + y = z, might have convinced Fermat that he had a proof of his last theorem. Also, in the light of Wiles’ proof of Fermat’s conjecture we show that there are only two sets of integer triples for all Diophantine equations of integer degree. Further, it is shown that unless two or more Diophantine equations are of the same degree (not necessarily an integer) then none of the triples of one set may be found in any of the others sets.
Table of Contents
1. Introduction
2. Diophantine equations, triples and Fermat’s last theorem.
3. Beyond integer triples
4. Discussion
Objectives and Topics
The primary objective of this work is to demonstrate, through a novel process involving the analysis of infinite sets of integer triples derived from Diophantine equations, that a specific structural dichotomy exists, which may provide a simplified perspective on Fermat’s Last Theorem and lead to a generalized conjecture regarding Diophantine solutions.
- Analysis of integer triples derived from Pythagoras’ equation and linear Diophantine equations.
- Investigation of the exclusivity of sets of triples across different degrees of Diophantine equations.
- Exploration of a potential heuristic path that Fermat might have utilized for his assertions.
- Formulation of "Fidler’s conjecture" regarding the non-overlap of triple sets for equations of differing degrees.
Excerpt from the Book
Pythagorean triples
Consider a pair, x, y, consisting of positive real numbers.
We may generate a triple, x, y, z , by inserting x and y into the Diophantine equation:
x^γ + y^γ = z^γ --------------------- (1).
Hence, z = (x^γ + y^γ)^1/γ ------------- (1a). If x, y and γ are specified then we have a triple which satisfies equation (1).
Now, we can generate an infinite number of triples by this process. However, we seek to investigate whether an equation of the form of (1) can be satisfied by a triple where all of the values are integers.
We posit, as will be demonstrated later, and on a temporary basis, that γ can be any positive integer equal to, or greater than, unity. Exactly as was done above, we now generate a set of triples for the equation when γ = 2. In passing, we note that z > x, y and it is essential for any proof that appears herein that this condition always obtains.
Hence, equation (1) becomes the widely-known equation called Pythagoras’ equation:
x^2 + y^2 = z^2 --------------------------- (2).
We can construct the RHS of this equation using any combination of the real numbers on the LHS. However we wish to investigate the triples of this equation when x, y, and z are integers. Such a triple is called a Pythagorean triple, and we denote this by the convention of naming the members of this set and other such sets, a, b , c , where, c > a, b.
Summary of Chapters
Introduction: This section provides a brief historical overview of Fermat’s Last Theorem and highlights the significance of Andrew Wiles' proof of the Taniyama-Shimura conjecture in modern mathematics.
Diophantine equations, triples and Fermat’s last theorem.: This chapter presents the core analysis of generating integer triples and establishes a contradiction when attempting to satisfy Diophantine equations of varying degrees using triples from different sets.
Beyond integer triples: The author extends the previous logic to real numbers and demonstrates that the fundamental dichotomy observed in integer triples holds, precluding the satisfaction of Fermat-type equations.
Discussion: This section synthesizes the findings into a general conjecture, named "Fidler’s conjecture," regarding the exclusivity of triple sets for Diophantine equations of identical degrees.
Keywords
Diophantine equations, Fermat’s Last Theorem, integer triples, Pythagorean triples, Wiles, Taniyama-Shimura conjecture, Fidler’s conjecture, degree of equations, mathematical proof, integer exponent, transcendental equations, number theory, triple sets, dichotomy, mathematics.
Frequently Asked Questions
What is the fundamental scope of this work?
This work explores the relationships between integer triples and Diophantine equations, aiming to show how these relationships can be used to analyze Fermat’s Last Theorem through a novel, set-based approach.
What are the central themes discussed in the paper?
The central themes include the generation of unique sets of integer triples, the mathematical constraints of Diophantine equations, and the theoretical exclusivity of solutions between equations of different degrees.
What is the primary objective or research question?
The research seeks to determine whether sets of integer triples derived from Diophantine equations of specific degrees can overlap, and if this analysis provides a logical basis for Fermat's assertions.
Which scientific methodology is employed?
The author uses a deductive analytical method, constructing infinite sets of triples and demonstrating a structural contradiction (a dichotomy) when attempting to map triples from one set into equations of a different degree.
What is covered in the main body of the text?
The main body details the mathematical formulation of Pythagorean triples, the extension to higher-order Diophantine equations, and the comparative analysis that leads to the rejection of mixed-triple solutions.
Which keywords best characterize this research?
Keywords such as Diophantine equations, Fermat’s Last Theorem, integer triples, and Fidler’s conjecture effectively capture the mathematical focus of the document.
How does the author relate his findings to the work of Andrew Wiles?
The author acknowledges Wiles' monumental proof of the Taniyama-Shimura conjecture and uses the findings therein to support the corollary that only two sets of integer triples exist in the range of sets satisfying Diophantine equations with an integer exponent.
What is "Fidler’s conjecture"?
Fidler’s conjecture states that unless a pair or more of Diophantine equations are of the same degree, none of the triples of numbers from one set can be found within the sets of triples of the other(s).
- Quote paper
- William Fidler (Author), 2020, Diophantine equations, triples and Fermat's last theorem, Munich, GRIN Verlag, https://www.grin.com/document/944286
