Re: FLTMA: A little group theory


The Dougster wrote:
Chip Eastham wrote:
The Dougster wrote:
The Dougster wrote:
The Dougster wrote:
Chip Eastham wrote:
The Dougster wrote:

Ah. The order of -1 is 2. | < -1 > | = 2. < -1 > = { -1, 1 }.

How do we know that if w^n == -1 mod z that
| < -1 > | divides | < w > | ?

Since <-1> is a subgroup of <w>, order of -1 (two) divides
the order of w.

Yipee! We're starting to use group theory to explore FLT!

http://www.mathpages.com/home/kmath264.htm

I think I see this more clearly today. If some power of w == -1 mod z
then, knowing w^0 = 1, we have { 1, -1 } <= <w> and so |<-1>| divides
|<w>|, where <= means "is a subgroup of".

I see in many sources on the web that without loss of generality,
certain conclusions may be made from a^n + b^n = c^n in Z. I have
concluded, with help here in sci.math, that exactly one of {x,y,z} is
even, and x < y < z < x+y. It might be more useful to give up x < y < z
< x+y and find instead that, say, y is even, as some web sources have.
I am still searching with Google for "Fermat's last theorem" and
"without loss of generality" OR "elementary". I want to get that stuff
out of the way, and certainly deduce as much as I can that might be
useful later.

Nearly a month now with no tobacco during the day, when I am out.

An equation I have seen in the elementary results is
x^p + y^p == x+y mod p, or something similar. That would make 4
equations in 4 unknowns.

Doug

I think it would be interesting to make a targeted search for
solutions with n = 3, the smallest possible prime, or even
better, to develop a proof that no such solutions exist.

Using what we have already shown, we need coprime x,y,z
such that:

(z/y) mod x has order 3
(z/x) mod y has order 3
(x/y) mod z has order 6 & (x/y)^3 == -1 mod z

where 0 < x < y < z < x+y and xyz even.

Thus phi(x) and phi(y) must be divisible by 3, and phi(z)
must be divisible by 6. Thus z = 7, 9, 13, 14, 19, 18,
etc. are candidates.

Chip
OK. I have done that in Mathcad, which doesn't share nicely in Usenet,
and am working in VB4 now, so I can do something up which can be
shared.

Is that xyz even or x+y+z even?

With x,y,z pairwise coprime, the notion that exactly one of x,y,z is
even is expressed equally well by xyz even or x+y+z even, but I
meant the former.

I wonder if I have access to a language in which operators can be
overloaded to make the source code look more like abstract algebra and
less like binary arithmetic. Hmm.

I like Prolog because one freely defines the relations/predicates that
make up the language (and they are readily "overloaded" for different
datatype, albeit at the expense of weak/runtime typing). However
you may want to look at some "functional" languages like Haskell
or ML. Lisp and descendants like Scheme are more "function"
friendly, but I don't know how important infix notation is for you.

Can anyone in sci.math suggest an appropriate limit for such a search?
Would 32-bit integers do all right? I think that's 16-bits of value
before the table a*b in Z/nZ* begins to overflow. There may be other
limits. Shall we say, to z<65535?

Yes, I think this would be plenty interesting, to run z out to 16 bit
unsigned limits, implemented in 32-bits for taking powers, etc.

regards, chip

.

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