Theorem 172 The equation x 4 y 4 z 4 has no solution in the integers except for

Theorem 172 the equation x 4 y 4 z 4 has no solution

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Theorem 17.2. The equation x 4 + y 4 = z 4 has no solution in the integers except for permutations of x = y = z = 0 or x = 0 , y = z = 1 . In the next lecture we will address the problem of which numbers can be written as the sum of two, three, and four squares. 58
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18. L ECTURE XVIII - F ERMAT S METHOD OF DESCENT One of the most important ideas in mathematics, which is most notably applied in number theory, is the idea of induction. A particular manifestation of this idea is Fermat’s method of descent. It capitalizes on a simple idea: there is no sequence a 1 , a 2 , a 3 , · · · of positive integers such that a i > a i +1 for all i . This follows from the well-ordering principle. The method of descent is a tool that can be used to complete a proof by contradiction. Suppose we wish to show that a certain equation has no positive integral solutions. We may suppose, for the sake of a contradiction, that a solution x 0 exists. Using x 0 , we construct a solution x 1 < x 0 . We continue to do this until we obtain an infinite chain of decreasing positive integral solutions, which would contradict the well-ordering principle. Fermat used his idea of descent to prove that the equation x 4 + y 4 = z 4 has no integral solu- tions except for the trivial ones (i.e. permutations of (0 , ± n, ± n ) and (0 , 0 , 0) ). Proof. We prove the stronger assertion that there are no triples ( x, y, z ) such that x 4 + y 4 = z 2 and xyz 6 = 0 . Suppose, for the sake of a contradiction, that there exists a triple ( x 0 , y 0 , z 0 ) of positive integers such that x 4 0 + y 4 0 = z 2 0 , and z 0 is smallest among all possible triples and z 0 > 1 . Note that such a z 0 exists by the well- ordering principle. Further, note that such a triple ( x 0 , y 0 , z 0 ) is necessarily primitive. Without loss of generality, assume that x 0 is odd. By Theorem 17.1, there exist odd integers m 0 , n 0 such that x 2 0 = m 2 0 - n 2 0 , y 2 0 = 2 m 0 n 0 , and z 0 = m 2 0 + n 2 0 . By primitivity, we must have gcd( m 0 , n 0 ) = 1 . Hence it follows that m 0 , n 0 , x 0 form a primitive Pythagorean triple, since x 2 0 + n 2 0 = m 2 0 . From here we see that n 0 is even, since x 0 is odd. From y 2 0 = 2 m 0 n 0 , it follows that one of m 0 , n 0 is an odd square and the other is twice an odd square. Hence m 0 = m 2 1 , n 0 = 2 n 2 1 with m 1 , n 1 odd. Then x 2 0 + 4 n 4 1 = m 4 1 , whence x 0 = (2 m 2 1 - n 2 1 )(2 m 2 1 + n 2 1 ) . Since ( x 0 , 2 n 2 1 , m 2 1 ) is a Pythagorean triple, there exist integers s 0 , t 0 such that x 0 = s 2 0 - t 2 0 , 2 n 2 1 = 2 s 0 t 0 , m 2 1 = s 2 0 + t 2 0 . It follows that s 0 , t 0 are squares, so there exist integers s 1 , t 1 such that s 0 = s 2 1 , t 0 = t 2 1 . Hence m 2 1 = s 4 1 + t 4 1 . However, z 0 > m 2 0 = m 4 1 , and this contradicts the fact that we assumed z 0 was the smallest. 59
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Fermat likely believed that the method of descent could be used to prove Fermat’s Last Theorem for every exponent n . This turns out not to be the case. However, the cases of n = 3 , 5 were done by Gauss and Euler respectively using different methods. Attempts to prove Fermat’s Last Theo- rem have lead to the discovery of many subjects in mathematics and number theory in particular, including all of modern algebra. Sir Andrew Wiles ultimately succeeded in proving Fermat’s Last Theorem in 1997 using methods from Iwasawa Theory, modular forms, and the theory of elliptic curves.
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