A Geometrical approach to Fermat's Conjecture

 

Escher, Concave and convex

The strategy of this proof is more elaborate than the preceding post but retains the key idea of analyzing the curve y = xⁿ in terms of the three variables a, b and c in the form of a diagram.The new proof introduces a model of Fermat’s Conjecture based on the implications of the anti-conjecture, the aim being to identify inherent contradictions.

The difficulty of the problem lies in the consistency of the equations derived from the hypothesis that the conjecture is false, the result being a series of tautologies that lead nowhere. Although the equations that arise appear to imply the irrationality of a key variable, it is usually the case that this condition cannot be established conclusively. In such cases the strategy is to reject the irrationality and follow up the implications of that decision. The situation is reminiscent of the Escher picture shown above, where everything seems to be consistent in two dimensions but must surely be false in a three dimensional world.

  

By extracting equations from the diagram, rather than just manipulating consistent equations derived algebraically, the inconsistencies in the assumptions are revealed and can be incorporated into a proof. This works because the geometric picture contradicts the equations derived from the defining equation aⁿ + bⁿ = cⁿ.

The details of the proof are appended below:

A Tangential Model of Fermat's Conjecture

Pierre de Fermat

My latest proof of Fermat's Conjecture is available for download here: Fermat Paper .A slide show video can be seen here FLT video The introduction and a few passages are listed below, subject to formatting limitations, together with the conclusions and explanatory diagrams. 

The main idea  is that a unique tangent to the curve xⁿ is determined by the variables a and c. The perpendicular drawn from the point where the tangent meets the cuve cuts the x-axis at K and it is the properties of K that determine the structure of the proof. The significance of this parametric variable is explained below in the passage about Pythagorean Triples.  

A key property of the curve, when n > 2, is that equal decrements to the left of K correspond to unequal increments to the right. This is because the differential coefficient of the curve is 

 nK^n-1  being non-linear but when n = 2, the differential coefficient is a linear function.

The proof strategy is to show that the variables a and c cannot both be integers when K is an integer. However, it is pretty clear that K is never an integer, so two further proofs are required under the assumptions that K is a rational fraction and K is an irrational number. 

When K is an integer it is impossible for the secant to pass through integer points in the plane. When K is a rational fraction, this is also the case. When K is irrational then the secant can pass through integer points but the variable b cannot be rational. This situation constitutes the proof of FLT. 

The mathematics involved in the proof is elementary and would certainly have been within the lexicon of Pierre de Fermat. An image of his 1621 translation of Diophantus whose margin could not hold Fermat's “marvelous proof” is shown below.

  

Introduction

The aim of this paper is to establish a more direct proof of Fermat’s Last Theorem than the proof published by Andrew Wiles in The Annals of Mathematics 142 (1995). The main idea is that the three terms aⁿ, bⁿ and cⁿ all lie on the curve xⁿ which facilitates the construction of diagrams showing the relation between these terms and several auxiliary variables used in the proof. The two diagrams used are shown in the annex.

Fermat’s Conjecture

Fermat’s Conjecture is: There are no natural numbers a, b, c, n such that aⁿ + bⁿ = cⁿ when n > 2.

FLT may be expressed formally as: NSabcn∈ ℕ aⁿ + bⁿ = cⁿ

subject to the following conditions:

(1)   a,b,c and n are distinct natural numbers               abcnℕ

(2)   a,b and c have no common factors                       abc NCF

(3)   n is greater than 2.                                     n>2

(4) a useful convention                                   a<b<c

(5) n is a prime number        

Parity limitations

Condition (2) implies that a, b and c cannot all be even numbers, otherwise they would have the common factor 2. Furthermore, if any two variables are even numbers then they have the common factor 2. Consequently, two of the factors must be odd and the third even. It will be shown later that b must be an odd variable given condition (3) so either a or c is even.

Analysis of K

This special characteristic explains why Pythagorean triples are possible because a and c always lie symmetrically about K. It will be shown below that this is not the case when n > 2  so the secant can never simultaneously intersect integer values of (a, aⁿ) and (c, cⁿ) when K∈ℕ.

The variables a and c can be defined in terms of K as follows:

a = K – q and c = K + p so that c – a = p + q, where p and q are deviations on either side of K.

A secant may be formed by moving the tangent an integer distance to the left of K.

When n =2 an equal deviation occurs to the right of K so that p = q. The underlying reason is that the differential coefficient of x² is the linear function 2x.  When n > 2  it is intuitively evident that an integer move to the left will result in a lesser and possibly fractional deviation to the right where        p < q.

Summary and conclusion

(1) If K is an integer then g is not an integer so either a is not an integer or c is not an integer.

(2) If K is a rational fraction then g cannot be an integer so either a is not an integer or

c is not an integer

(3) If K is an irrational number then either n is irrational or b is irrational.

In arriving at these conclusions the negative hypothesis Sabcn∈ℕ aⁿ + bⁿ = cⁿ was assumed together with the conditions (1) through (5) applicable to FLT. The three conclusions above prove that the negative hypothesis under the given conditions is inconsistent, consequently FLT

must be true reductio ad absurdum.