Proving Fermat's Last Theorem for N = 4
Date: 05/18/2000 at 09:03:55 From: Henryk Dabrowski Subject: Fermat's Last Theorem I don't know how to prove Fermat's Last Theorem for the specific case n = 4. Can you help me?
Date: 05/18/2000 at 14:51:07 From: Doctor Rob Subject: Re: Fermat's Last Theorem Thanks for writing to Ask Dr. Math, Henryk. To show that x^4 + y^4 = z^4 is impossible, we actually prove the stronger fact that x^4 + y^4 = w^2 is impossible, whether or not w is a perfect square, except when x*y = 0. First we restrict our attention (without loss of generality) to the case GCD(x,y) = 1. Assume that there is a solution (x,y,w) with x > 0 and y > 0. Among all of them, pick the one with the smallest w > 0 (this exists because every nonempty subset of the natural numbers has a smallest element). Then (x^2,y^2,w) is a primitive Pythagorean triple. Assume (without loss of generality) that y^2 is the even member of the triple. Then there exist two integers u and v with: u > v > 0, GCD(u,v) = 1, and u + v odd, such that: x^2 = u^2 - v^2, y^2 = 2*u*v, and w = u^2 + v^2. If u were even, then v would be odd, and x^2 = 0 - 1 = 3 (mod 4), which is impossible. Thus u is odd and v is even. Then (y/2)^2 = u*(v/2), GCD(u,v/2) = 1. This implies that for some integers r and s, u = r^2, v/2 = s^2, y = 2*r*s, GCD(r,s) = 1, r > 0, s > 0, and r is odd. Also x^2 + v^2 = u^2, so we have x^2 + 4*s^4 = r^4. Since GCD(r,2*s) = 1, we have another primitive Pythagorean triple (x,2*s^2,r^2). Thus there are integers a and b such that x = a^2 - b^2, 2*s^2 = 2*a*b, r^2 = a^2 + b^2, GCD(a,b) = 1, a > b > 0, and a + b odd. Since a*b = s^2, we can write a = f^2, b = g^2 with some f > 0 and g > 0 with GCD(f,g) = 1. Then we have r^2 = f^4 + g^4, which looks just like our original equation. Now w = u^2 + v^2 = r^4 + 4*s^4 > r^4 >= r > 0, so w > r > 0 and (f,g,r) is a different solution from (x,y,w), and its last component is smaller. This is a contradiction. Thus our assumption that a solution (x,y,w) exists with x > 0 and y > 0 must be incorrect. This means that the equation x^4 + y^4 = w^2 has no integer solution unless x*y = 0. This implies Fermat's Last Theorem for the case n = 4. This is a classic application of Fermat's Method of Infinite Descent. In fact, it is a proof essentially due to Fermat himself. I have adapted the proof from Ivan Niven and Herbert S. Zuckerman, _An Introduction to the Theory of Numbers_ (1960), pp. 100-102. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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