FLT simple question
From: richard miller (richard_at_microscitech.freeserve.co.uk)
Date: 08/11/04
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Date: Wed, 11 Aug 2004 18:52:35 +0100
!NOT A CRANKY PROOF CLAIM!
I'm trying to find a reference to the following point which maybe true or
false
Start
IF the integer triple (a,b,c), a<b<c, co-prime in pairs, usual FLT
restrictions, were to be an FLT counter-example, odd prime exponent n, such
that
a^n+b^n=c^n
then a and b have to be of the form, integers x,y,k,l>=1
1) a = x(2kn+1)
(a is prime or composite)
2kn+1 is prime, x is composite or prime, x>=1
2) b = y(2ln+1)
(b is composite)
2ln+1 is prime, y is composite or prime, y>1
End
The condition (1) is actually relatively simple and no big deal. The '2ln+1'
bit comes from Lagrange's Theorem. Condition (2) follows by symmetry
(almost) in a and b with respect to c.
If you're wondering, it comes from another FLT-like, Diophantine Equation I
am currently looking at for which the assertion made above is a condition
for solutions.
As an illustration, for the cubic case n=3, an almost plausible triple would
be (31,38,39) where x=1, k=5, y=2, l=3. Of course, they are all doomed to
failure!
Nearly all the literature on FLT and 'FLT-like' Diophantine equations
concentrates on the exponent. Browsing the Web, Papers and books is just
overload when using FLT as the search key. A pointer would be helpful.
Thanks in anticipation
Richard Miller
[ Moderator's note: yes, we know that since FLT is true all statements
of the form "if X is a counterexample to FLT then Y" are true. The
question is, what can be proven without using FLT?
-ri ]
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