Fermat's Last TheoremDate: 02/02/2002 at 00:23:42 From: blaine shaw Subject: Fermat's last theorem Proof of Fermat's last theorem... Date: 02/02/2002 at 05:48:17 From: Doctor Anthony Subject: Re: Fermat's last theorem The actual proof is about 200 pages of very advanced mathematics that only specialists are likely to understand. Below is an outline of the proof. Fermat's Last Theorem states that there are no integer solutions of the equation x^n + y^n = z^n if n > 2 i.e. there are no integers x, y, z such that x^3 + y^3 = z^3 or integers x, y, z such that x^7 + y^7 = z^7. This is easily stated but has proved to be one of the most vexing problems in the whole history of mathematics. Fermat, (1601-65), claimed to have a 'marvellous' proof which the margin of his book was 'too small to contain.' For the next 350 years mathematicians tried in vain to find a proof. Indeed, some mathematicians devoted much of their life's work to the pursuit of that goal, and the search for a proof led to the development of whole new branches of mathematics, but it was only in the last decade of the 20th century that Andrew Wiles finally completed the task. The actual proof is very indirect, and involves two branches of mathematics, which at face value appear to have nothing to do either with each other or with Fermat's theorem. The two subjects are elliptic curves and modular forms. Below is a brief description of what these are, and how they are related to Fermat's Last Theorem. Elliptic curves are of the form y^2 = x^3 + ax^2 + bx + c where a, b, c are integers. The problem with elliptic curves is to find if they have integer solutions, and if so, how many. For example the equation y^2 = x^3 - 2 with a = b = 0 and c = -2 has only one set of integer solutions, namely x = 3, y = 5, but proving that there are no other solutions is extremely difficult. The problem is simplified by making the possible numbers finite, i.e. working in 'clock' arithmetic. So 5-clock arithmetic uses only 0, 1, 2, 3, 4 then 5 = 0 again. (You may recognize this as 5 congruent 0 mod(5)). It was then possible to make progress with determining the number of integer solutions of the elliptic curves. For a particular elliptic curve, the number of integer solutions in each clock arithmetic forms an L-series for that curve. Example: Elliptic curve x^3 - x^2 = y^2 + y L-series L1 = 1 number of solutions in 1-clock arithmetic L2 = 4 " 2-clock " L3 = 4 " 3-clock " L4 = 8 " 4-clock " L5 = 4 L6 = 16 L7 = 9 L8 = 16 ....... ....... This series can go on as far as you like. Because we cannot say how many solutions there are in normal number space, extending to infinity as it does, the L-series gives a great deal of information about the elliptic curve it describes. The idea is that studying the L-series you can learn all you want to know about its elliptic curve. A modular form is defined by two axes, x and y, but EACH axis has a real and an imaginary part. In effect it is four-dimensional (xr, xi, yr, yi) where xr means real part of x, xi means imaginary part of x, and similarly with yr and yi. The four-dimensional space is called hyperbolic space. The interesting thing about modular forms is that they exhibit infinite symmetry under transformations of the type az+b f(z) -> f[------] cz+d These are functions that remain unchanged when the complex variable z is changed according to the above transformation. Here the elements a, b, c, d, arranged as a matrix, form an algebraic group. There are infinitely many possible variations. They all commute with each other and the function f is invariant under the group of transformations. Modular forms come in various shapes and sizes, but each one is built from the same basic ingredients. What differentiates each modular form is the amount of each ingredient it contains. The ingredients of a modular form are labelled from one to infinity (M1,M2,M3,....) and a particular modular form might contain one lot of ingredient one (M1=1), three lots of ingredient two (M2=3), two lots of ingredient three (M3=2) and so on. So now we get an M-series M1 = 1 M2 = 3 M3 = 2 ...... ...... and so on. Note about INGREDIENTS The mathematics of modular forms is very abstract and difficult to express in terms that are familiar to any but a specialist in the subject. If you have studied matrix theory you will know that a transformation matrix will have associated eigenvalues and eigenvectors. The eigenvector is a vector which remains invariant (except perhaps in magnitude) under the operation of the transformation matrix. The eigenvectors are used amongst other things in diagonalizing a matrix, that is expressing a matrix in a form where any power of the matrix can be calculated in a few seconds. The 'ingredient' that we mention is a property of these eigenvectors and what are called Hecke operators. Below is a brief definition of these but I won't promise that you will feel much the wiser after reading it. Hecke Operators The space of modular forms M = M(N) is finite dimensional. There are special operators called Hecke operators (denoted by T(p), one for each prime p) on this space. M is simultaneously diagonalizable by the T(p): it has a basis of modular forms that are eigenvectors for each T(p). At this point we come to the work of Taniyama and Shimura, who found a strange affinity between some elliptic curves and some modular forms. However far you took the L-series and the M-series for a particular elliptic curve and a particular modular form the two matched exactly. This led to the Taniyama-Shimura conjecture that ALL elliptic curves are modular. It was in proving this conjecture that Andrew Wiles established the proof of Fermat's Last Theorem. The reason they are connected is as follows. Gerhard Frey showed that IF there was a solution in integers to x^n + y^n = z^n, say A^n + B^n = C^n then we could get an elliptic curve of the form y^2 = x^3 + (A^n-B^n)x^2 - A^n.B^n Another mathematician, Ken Ribet, showed that this equation could not be modular. So now we have the following chain of reasoning: 1) If the Taniyama-Shimura conjecture can be proved, then every elliptic curve is modular 2) If every elliptic curve must be modular, then the Frey elliptic curve is forbidden to exist. 3) If the the Frey elliptic curve does not exist, then there can be no solutions to the Fermat equation. 4) Therefore Fermat's Last Theorem is true. The greatest difficulty was in proving that the Taniyama-Shimura conjecture was true. This is the contribution made by Andrew Wiles, and the final stage in establishing Fermat's Last theorem. - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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