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Fermat's Last Theorem with Negative Exponents


Date: 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/   
    
Associated Topics:
High School Basic Algebra
High School Number Theory

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