From: ksbrown@ksbrown.seanet.com (Kevin Brown)
Newsgroups: sci.math
Subject: Re: Fermat's Theorum and the Simpsons
Date: Tue, 07 Nov 1995 06:01:00 GMT

Orac (100124.3523@compuserve.com) wrote:
> Is 1782^12 + 1841^12 = 1922^12 the nearest to disproving Fermat?
 
I'm curious about how this particular example was selected.  Of
course, as has been mentioned, it's easy to see that the "equality"
can't hold.  For example, it doesn't satisfy odd/even parity, and 
the equality is ruled out by the elementary proofs for exponents 4 
and 3.  Also, the absolute error is 700212234530608691501223040959,
which doesn't compare very well with "near counter-examples" such as

                3086^3 + 21588^3 = 21609^3

which has an absolute error of just 1.  In fact, it's easy to
construct infinitely many examples with an error of 1 by means 
of the identities

      (9u^3 + 1)^3  +    (9u^4)^3    =  (9u^4 + 3u)^3  +  1

      (9u^3 - 1)^3  +  (9u^4 - 3u)^3  =  (9u^4)^3  -  1

In view of these examples, it's tempting to think the Simpson
example was just naively constructed by arbitrarily choosing 
the exponent 12, picking the first two four-digit numbers that 
come to mind, and then computing the remaining base (approximately).

However, this doesn't usually work.  For example, if you choose, say,
1789 and 1914 for the left hand bases (with exponent 12), then the
right hand base comes out as something like 1973.5837...  In contrast,

using Simpson's values of 1782 and 1841, the right hand base comes 
out to be 1921.99999995586...  This suggests the example is not
totally naive, and must have been constructed by some non-trivial
method.  

I remember reading an article once about constructing arbitrarily 
close "counter-examples" to FLT for any given exponent (in the sense
that (a/c)^n + (b/c)^n  is arbitrarily close to 1).  Does anyone 
recall the details?  Is Simpson's example taken from the article?