Shi Feng Sheng Danny Wong

Transcription

1 Exhibit C A Proof of the Fermat s Last Theorem Shi Feng Sheng Danny Wong Abstract: Prior to the Diophantine geometry, number theory (or arithmetic) was to study the patterns of the numbers and elementary calculation techniques The 1637 Fermat s Last Theorem (FLT) is the 1 st important elusive problem in number theory originated from Diophantine geometry. Leonhard Euler tried and failed to resolve the (FLT) during the 18 th century. However, it marked the (rebirth) of (FLT) as the beginning of modern number theory. (FLT) has resisted every attempt at a solution, and may very well hold the key to the overall understanding of the problems in number theory. After the proclaimed proof the Goldbach s conjecture, it became obvious that (FLT) is not an enigma. The objective here is to illustrate that Fermat s Last Theorem can be resolved by a relatively simple non-geometry-based procedure. 1. The Problem In about 1637, French mathematician Pierre de Fermat ( ) noted in his copy of Arthimetica that he had a remarkable proof (by descent) that this indeterminate Diophantine equation is not solvable when N is an integer > 2. Fermat further noted that there was very little space left in the margin of his book to elaborate. 2. Early History Euclid s Elements (300B.C) introduced Euclidean geometry and provided the 1 st proof of infinitude of primes by abstract reasoning. Elements also devoted a part to primes and divisibility, topics that belong unambiguously to number theory. Diophantine (3 rd century) introduced a series book called Arithmetica. Diophantine studied rational points on the curves and algebraic varieties, and showed how to obtain infinitely many rational (points) satisfying a system of equations by giving a procedure that can be made of algebraic expression (the basic algebraic geometry in pure algebraic terms). Diophantine was the first mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficient and solutions. Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought. Unfortunately, most of the texts are lost. Nonetheless, Diophantine geometry and of Diophantine approximations are important to the researchers today. After the Greeks, not much happened to primes until Fermat introduced his prime numbers and number theory in the 17 th century. Unfortunately, his entire work on number theory was in the forms of private letters and marginal notes, and Fermat s Last Theorem was found long after his death without elaboration. (FLT) led to tremendous advances in number theory, and number theorists studied primes and the properties of objects make out of integers (such as rational numbers) or defined as generalization of integers (algebraic integers). Integers can be considered either in themselves of as solutions to equations (Diophantine geometry). 1 / 5

2 Moreover, the analytic geometry (coordinate geometry) was introduced by Rene Descartes ( ) and Fermat ( ). Fermat primarily studied the properties of elliptic curves (Diophantine equations). Descartes primarily studied the classic work on conic and cube; he introduced rectangular coordinates to locate points and enable lines and cures to be represented with algebraic equations, his algebraic geometry extended subject to multidimensional and non-euclidean spaces It was almost a century later, the analytic geometry was accepted by the 18 th century mathematicians mainly because it supplied with concrete quantitative tools needed to to study physical problem 1 using calculus of Newton and Leibniz. And during the same period; (a) Leonhard Euler ( ) became interested of number theory after worked on many Fermat s unelaborated problems, including his unsuccessful attempt to resolve the (FLT), (b) Euler also applied the technique of the infinitesimal calculus and zeta function to investigate prime distribution, but unsuccessful in all front. Nonetheless, his failed attempt to resolve the (FLT) marked the rebirth of (FLT) as the beginning of modern number theory. 3. Recent History In about , Dr. Andrew Wiles proclaimed a lengthy proof of (FLT), which was based on a certain elliptic curve (from the 1954 Taniyama conjecture) did not exist and other deep theories. However, no one agreed that his proof was what Fermat had in mind due to the fact that his modern, far-reaching vision of mathematics did not exist during the Fermat s era. And during the same period, the 1993 Beal s conjecture was coined. Mr. Beal believed Fermat had a relatively simple non-geometry-based proof of (FLT). Beal s conjecture was accepted as an open question in number theory because his approach via common factor was very close to the research activities of the professionals. 4. Observation Fermat developed methods for (doing what in today s terms) amount to finding points on the elliptic curves (genus 1), but his annotation was completely divorced from the method of Diophantine. Fermat must have a simple non-geometric-based method due to the fact that: 1. Fermat claimed that his proof would fit into the margin of a book if he had a little more space; which implied that his proof (by descent) must be a very short one. 2. Fermat also implied that has solution when N = (an integer N > 2) is equivalent to (N 3), and Fermat never constrained X or Y in X N + Y N = Z N, so the options are: X = Y or X Y. 4. The proof of (FLT) by Dr. Wiles was based on deep modern theories that did not exist during the Fermat days. 5. Coincided with Mr. Beal s point of view. 1 Historically, mathematician/physicist Daniel Bernoulli ( ) was at the forefront to adapt infinitesimal calculus to analyze his physical fluid motion problem, which his good friend Leonhard Euler ( ) also contributed greatly. 2 / 5

3 4. The Solution The proclaimed proof of Goldbach s conjecture illustrated that all the even numbers > 2 can be expressed as the sum of two prime numbers, obviously, 4 = (A) Fermat claimed that he had a proof (by decent) that equation is not solvable when N is an integer > 2. Since (N > 2) is equivalent to (N 3). Let: N descends from 3 to (1 and 2): Let: X = Y = 2 N = 1, 2, 3 N = 1: we have: = 4 N = 2: X N + Y N Z N ( ) The same argument can be constructed to hold false for all the exponent N > 2. So: N = 3: X N + Y N Z N ( ) N = 4: X N + Y N Z N ( ) N = 5: X N + Y N Z N ( ) Let: X Y, so X = 2, Y = 3 N = 1, 2, 3 N = 1: we have: = 5 N = 2: X N + Y N Z N ( ) The same argument can be constructed to hold false for all the exponent N > 2. So: N = 3: X N + Y N Z N : ( ) N = 4: X N + Y N Z N ( ) N = 5: X N + Y N Z N ( ) Fermat is correct that he had a remarkable proof (by descent) that this indeterminate Diophantine equation is not solvable when N is an integer > 2. Moreover, according to the Pythagorean Theorem: X + Y = Z (X + Y) 2 = Z 2 X 2 + 2X.Y + Y 2 = Z 2 X 2 + Y 2 = Z 2-2XY So: X 2 + Y 2 Z 2 3 / 5

4 (B). Fermat only claimed that has no solution when N is an integer > 2, which implied that has solution when N = 2: When N = 2: we have: X 2 + Y 2 = Z 2 From this perspective, Fermat could only mean the formula of Pythagorean right triangle: Let X = Y: (X + Y) 2 = Z (1/2 XY) X 2 + 2XY + Y 2 = Z 2 + 2XY X 2 + Y 2 = Z 2 + (2XY 2XY) X 2 + Y 2 = Z 2 5. Additional Support Not everyone will accept this elementary non-geometry-based proof, but it is what it is. Moreover, additional [studies] further supported that this proof of the (FLT) is correct: [Study a]: Integer N > 2 is same as N 3, and Fermat never constrained X and Y. So: Let: X = Y = 2, descending N from 3 2: Variation: X N = Z N Y N When N = 2: 2 2 = Z = Z 2 4 Z 2 = 8 Obviously: X N = Z N Y N can not be equal. However: Z 2 = 8 = 2 3 So: 2 2 = Variation: = 2 3 (4 + 4 = 8) An algebraic equation (of a particular form) can be formed so its solution to one N can be adapted to solve all the others: 2 N + 2 N = 2 N+1 N =1, 2, 3 N = 1: = = 4 N = 2: = = 8 N = 3: = = 16 N = 9: = = 2048 It seems unnecessary to continue any further. Fermat is correct that this indeterminate equation has no solution when N is an integer > 2. In fact, the equation X 2 + Y 2 = Z 2 could only exist as the formula for Pythagorean right triangle.. 4 / 5

5 [Study b]: Log X N = N log X, Log X + Log Y = Log (X x Y) Let: N = 2, 3, 4 Log (X N + Y N ) = N log Z log ( X N + Y N ) N = = 1. (A) log Z Log 10 = 1; therefore, the only possibility for N to have a whole number solution in equation (A) is when X N + Y N = 10 and Z = 10: Take: Variation: X N + Y N = 10 N X N = 10 N Y N N log X = log (10 N Y N ) log (10 N Y N ) But: N = (B) log X Log 10 = 1; therefore, it is impossible for equation (B) to have any whole number solutions. Fermat is correct that this indeterminate equation has no solution when N is an integer > The Conclusion (FLT) is correct, and it is well founded that Fermat claimed that he had a truly remarkable proof by (descent) due to the fact that this indeterminate equation X 2 + Y 2 = Z 2 can not be solved as an algebraic equation, but the answer could be verified. External links are retrieved for general information related to Fermat s Last Theorem only: [1]. The world s most famous math problems by Marilyn Vos Savant. [2]. a generalization of Fermat s Last Theorem: The Beal s conjecture and prize problem, by Daniel Mauldin. [3]. [4]. [5]. [6]: geometry 5 / 5