Fermat's Last Theorem with Negative ExponentsDate: 10/26/2000 at 13:49:22 From: Anonymous Subject: Fermat's Last Theorem for n < 2 Fermat's Last Theorem has been proven for n > 2, but has anyone investigated n < 2? That is, are there any solutions of x^n + y^n = z^n for n < 2? Obviously n = 1 and n = -1 are trivial, and n = 0 leads to 1+1 = 1, so there are no solutions to that. I haven't found any solutions for n = -2, etc. by trial and error. Has anyone saved me the trouble of investigating further by proving the existence or non-existence of these solutions? Date: 10/27/2000 at 03:12:51 From: Doctor Floor Subject: Re: Fermat's Last Theorem for n < 2 Hi, Thanks for writing. The solutions for n are more or less equivalent to the solutions for their opposites, -n. Suppose we have a solution, for some n x^(-n) + y^(-n) = z^(-n) Multiply by (xyz)^n and you get (yz)^n + (xz)^n = (xy)^n. This shows that if there is a solution for x^n + y^n = z^n for a certain n, then also for -n. And thus there can't be solutions for n < -2. If you have more questions, just write back. Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ |
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