Fermat's TheoremDate: 01/21/98 at 01:08:30 From: Vincent M. Li Subject: Fermat's Theorem I have heard a lot about Fermat's Theorem, and how it was once considered one of the world's greatest unsolved mathematical mystries. Supposedly it has been solved by a Professor Andrew Wiles. I was just wondering why it was such a mystey, and how it was proved. I probably won't understand the implications of this for the known world but I am curious to learn. Date: 01/21/98 at 15:20:02 From: Doctor Wilkinson Subject: Re: Fermat's Theorem Here's a webpage with a lot of information: http://www.best.com/~cgd/home/flt/flt01.htm -Doctor Wilkinson, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 01/21/98 at 15:40:21 From: Doctor Rob Subject: Re: Fermat's Theorem You can try the following URL for some of the history of this problem: http://daisy.uwaterloo.ca/~alopez-o/math-faq/node22.html Also go to our Archives search form: http://mathforum.org/mathgrepform.html and search for "Wiles". This will give you a lot of information on this subject. It was such a mystery because Fermat not only stated the problem, but claimed he had a solution. 350 years passed with many of the greatest mathematicians straining to rediscover that solution, but failing. It is now thought that Fermat was mistaken. Clearly the solution by Wiles is not the one Fermat thought he had - he could not have known even a fraction of what Wiles used in his proof. It is also a mystery because it is so easy to state that any student of beginning algebra can understand the problem, but the proof is so extremely difficult. The first step in learning the kind of math you would need to understand Wiles's proof is to learn about curves in the plane. You already may be familiar with those of degree 2, the ellipse, parabola, and hyperbola. The study of curves of degree 3 is the jumping-off point for this subject. Most such curves are called "elliptic curves" because of their connection will elliptic functions and elliptic integrals. Elliptic integrals are so named because you need to evaluate one to find the arc length of an ellipse. As you know, elliptic integrals cannot be expressed in closed form in terms of the usual functions of mathematics. That is where the elliptic functions arise. One elliptic function, the Weierstrass P-function, and its derivative P' satisfy an equation of degree 3, called an elliptic curve. The particular elliptic curve Wiles used is y^2 = x*(x + a^n)*(x - b^n), where a and b are positive integers, and n is a prime number >= 5. This is how we get a connection to Fermat's equation a^n + b^n = c^n. Elliptic curves have many wonderful and surprising properties. For one thing, the points on the curve with a certain operation form an abelian group. The interplay between the abstract algebra (groups), complex analysis (elliptic functions), and geometry (curves) leads to some very powerful and exciting mathematics. Elliptic curves seem to have applications to a surprisingly diverse set of areas of mathematics, such as solving Diophantine equations, sphere packing, and proving large integers prime. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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