Fermat's Last Theorem and EulerDate: 11/29/2004 at 21:45:39 From: Darren Subject: x^3+y^3=z^3 I was reading through different sites about Fermat's Last Theorem, and I remember reading that Euler showed the case x^3 + y^3 = z^3 had no integer solutions easily--the key word being easily. I haven't been able to find any site that explains how to show this. Date: 11/30/2004 at 02:33:12 From: Doctor Nitrogen Subject: Re: x^3+y^3=z^3 Hi, Darren. You will find a proof of an important related proposition offline in chapter 17, Section 8, pages 284, 285, of: [1] "A Classical Introduction to Modern Number Theory," Second Edition, Kenneth Ireland, Nathan Rosen, Springer-Verlag, ISBN# 0-387-97329-X. The proposition proved in this book is: Proposition: The equation x^3 + y^3 = u*z^3, where u is a fixed unit in Z[omega], has no integral solution (x, y, z), xyz =/= 0, where x, y, z are elements of Z[omega]. The immediate implication is that, and I am quoting from page 284 here: "...a nonzero cube in Z [i.e., Z is the ring of integers] is not the sum of two nonzero cubes in Z." In Chapter 1, Section 4, pages 12-13, Z[omega] is defined as being this commutative ring below: Z[omega] := {a + b*omega | a, b are in Z}, and where omega = -1 + (sqrt(-3))/2, is one of the 3rd roots of unity. In Chapter 1, Section 4 of [1], Z[omega] is shown to be an integral domain as well as a Euclidean domain. The proposition is important in proving FLT when the exponent n = 3. Incidentally, that x^p + y^p + z^p = 0, where p is a prime > 2, has no integer solutions with xyz =/= 0, is related to work by Taniyama, Shimura, Wiles, Ribet and Frey on elliptic curves, which are also sometimes called algebraic varieties, under suitable conditions. Hope this helped answer the questions you had concerning your mathematics problem. Feel free to write back if you have further questions. - Doctor Nitrogen, The Math Forum http://mathforum.org/dr.math/ |
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