Re: FLTMA: A little group theory


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?

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.

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?

Doug

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Relevant Pages

  • Re: FLTMA: A little group theory
    ... Yipee! ... We're starting to use group theory to explore FLT! ... < x+y and find instead that, say, y is even, as some web sources have. ... I think it would be interesting to make a targeted search for ...
    (sci.math)
  • Re: FLTMA: A little group theory
    ... The Dougster wrote: ... Yipee! ... We're starting to use group theory to explore FLT! ... < x+y and find instead that, say, y is even, as some web sources have. ...
    (sci.math)
  • Re: FLTMA: A little group theory
    ... Chip Eastham wrote: ... Yipee! ... We're starting to use group theory to explore FLT! ...
    (sci.math)