Re: FLTMA: Using FLT as a lemma
- From: "The Dougster" <DGoncz@xxxxxxx>
- Date: 11 Aug 2006 01:17:04 -0700
If n below is never odd prime, it is always comppsite or 2. If it is
always composite, then either at least one pair of the sub-sequence
periods share no common factor or all three are degenerate with period
2. Why would *that* be true, given conditions 1), 2), and 3)? Are there
any other conditions deduced from FLT that apply that might give
insight as to why this common factor is absent?
Doug
The Dougster (I) wrote, in the OP on this lonely thread:
If there were a counterexample, integers {n,a,b,c} to FLT, n>2,
a^n + b^n = c^n (and there isn't),
then there would be a primitive counterexample {p,x,y,z},
x^p + y^p = z^p with:
1) gcd(x,y,z) = 1
2) exactly one of {a,b,c} even
3) 0 < x < y < z < (x+y)
4) p an odd prime
For this nonexistent counterexample,
(x^n + y^n) == 0 mod z,
(z^n - x^n) == 0 mod y, and
(z^n - y^n) == 0 mod x.
For any positive x, y, and z, under conditions 1), 2), and 3,
a sequence S(n) exists, with mod indicating the remainder of division:
S(n) =
(x^n + y^n mod z +
(z^n - x^n) mod y +
(z^n - y^n) mod x;
S(0) = 2.
Now, since FLT is true, we know the zeros of this sequence will never
be at
any odd prime.
There might be something learned by proving n is not
an odd prime without using FLT. Other results might be
had using FLT. Much is made in the popular press of the proof of FLT,
but I
haven't heard of it being applied to *do* something. But then, I don't
read
the literature of mathematics.
Has FLT been used as a lemma in any work you know about?
I have Fermat's Last Theorem for Amateurs by Paulo Ribenboim, which is
not
much help, and a bound print of a thesis by Amy Glen at the University
of
Adelaide in Australia, which has been interesting reading.
Doug
.
- References:
- FLTMA: Using FLT as a lemma
- From: The Dougster
- FLTMA: Using FLT as a lemma
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