Re: -- FLTMA and the existing proof

On Sat, 26 Jan 2008 02:11:24 -0800 (PST), The Dougster 22044
<DGoncz@xxxxxxxxxxxx> wrote:

Well, then I am done. Four years or so tinkering with this has
produced this last thread in which conditions were given one by one
and counterexamples given at each stage. I am out of conditions but
only one counterexample is needed.

It's not hard to repair your conjecture, at least enough to get it
back on the road. No warranty though.

The quick fix is simple ...

Dougster's conjecture (latest revised version):

There do not exist positive integers x,y,z such that

(1) x < y < z < x+y

(2) x,y,z, are pairwise coprime

(3) For some prime p > 2,

x^2 | z^p - y^p
y^2 | z^p - x^p
z^2 | x^p + y^p

(4) p | (x + y - z)

(5) x,y,z are not all odd.

quasi

P.S. (1) Sorry, for your loss.

P.S. (2) Don't give up your number theory ideas. Counterexamples
notwithstanding, you've come up with some cool insights, leading to
some fun, interesting explorations. My suggestion is to temporarily
table the FLT-related ideas, and instead start a rigorous self-study
of ...

(1) Number Theory -- just divisibility and congruences,
but with mastery and full rigor.

(2) Abstract Algebra -- groups, rings, fields

(3) Linear Algebra -- a quick review of the basics
(vector spaces, bases, dimension, linear maps)

(4) Galois Theory -- continuation of field theory

(5) Algebraic Number Theory

(6) Computational Algebra & Number Theory

There are further directions to take, but I think the above subjects
would make for a good start. It's ok to do more than one subject at a
time. Also, the order suggested above is not cast in stone.

Try to choose just the right books. Select texts which are at the
right level for you -- not too elementary, not too advanced (although
a little too advanced is sometimes ok). Make sure (by trying a
chapter) that the text is well written (from your perspective), and
has interesting, insightful exercises at various levels of difficulty.

If possible, try to defer the FLT investigations for now, using the
time instead to master those subjects which can provide more powerful
tools for future investigations. As a minimum, you'll gain more
sophisticated insights.

Good luck.

quasi
.

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