From: batchbiz@aol.com (BatchBiz)
Newsgroups: sci.math
Subject: Alternative form of FLT
Date: 17 Sep 1998 15:29:05 GMT

The equation a^m+b^m=c^n has a non-trivial integer solution if and onty if
(m,n)=1.
Proof: If (m,n)=1 there exist x,y such that xm+1=yn. Then 2^xm+2^xm =2^(xm+1)
  =2^ym. (there are other solutions)
  If (m,n) >2 FLT shows there are no solutions.
  If (m,n) =2 there are no solutions: a^2m+b^2m=c^2 has no solutions; the
proof follows the lines of that for a^4=b^4=c^2
  I think this is cool:-
  1.  It seems strange that such a simple formula provides so many
    solutions.  When the formula fails there are no solutions.
  2.  It seems strange that FLT zeros in on the cases where there is no 
   solution.  In all cases where FLT does not forbid it there is a solution.
  3.  "If and only if" is cool.
  4.  A novice tends to think (I did)that FLT is probably true for large n 
     just because there are so few n-the powers that it is unlikely that two
     will happen to add to another.  In this form, this thought is clearly
WRONG!

  This result is (surely) known, but I haven't seen any reference in any text.
  What is its history? Does it lead anywhere?
     
     Dick Batchelor
==============================================================================

From: batchbiz@aol.com (BatchBiz)
Newsgroups: sci.math
Subject: Re: Alternative form of FLT
Date: 18 Sep 1998 13:11:59 GMT

>=> The equation a^m+b^m=c^n has a non-trivial integer solution if and onty if
>=> (m,n)=1.
>
>Say, what? What if m = n = 2?
>
Thank you. I meant to say 
  "(m,n)=1 or m=2"
  Dick Batchelor