I read that proving Beal would also prove FLT as FLT is a special
case of Beal. I don't see how. Aside from the structure of the equations
in question (Beal: a^x+b^y=c^z, FLT: a^n+b^n=c^n) the two statements are
different. The FLT equations could be said to form a subset of the Beal
equations but Beal says that the solution triplets are never relatively
prime, whereas FLT says there exist no solutions!
My counter example program focused on the case where a<>b and b<>c. I was
not intersted in the a=b=c case as these solutions would have an automatic
GCF>1 (nontrivial common factor = a = b = c). Also, to save computing
time, I restricted the exponents in like manner: x<>y and y<>z. If we
accept Dr. Wiles' proof as correct, then the Beal Conjecture does not
apply when x=y=z. The program found no counter examples, but all the
solutions had GCFs equal to 2 or 3. There must exist larger GCFs, but my
computer can only represent exact integer values up to 2^64-1 and these
powers can get pretty big pretty fast! I must construct an integer type
with maxint=10^100 (googol) and try again.
Regards
On Sat, 29 Nov 1997, mark snyder wrote:
> At 4:03 PM -0500 11/29/97, Alvar J. Garcia wrote:
> >
> > Beal's Conjecture may well not be as compelling as FLT, but it is of
> > interest.
> >
> > Think of the Pythagorean case. a^2+b^2=c^2 which has infinitely many
> > solutions - Pythagorean Triplets - where a,b,c are unequal positive
> > integers. Of these triplets, some are mutually prime (no GCF>1) such as
> > 3,4,5 and some are not such as 10,24,26. Upon closer inspection, however,
> > it would seem that those that are not mutually prime are simply multiples
> > of those that are (10,24,26 = 2*(5,12,13)). Is the fact that there are
> > infinitely many Pythagorean Triplets due to these mutiples? More to the
> > point, are there infintely many mutually prime Pythaorean Triplets?
>
> Every Pythagorean triple is a multiple of a *primitive* Pythagorean triple,
> where a PPT is a Pythagorean triple (a,b,c) with GCD(a,b,c) = 1 (Proof: if
> there is a GCD, factor it out of each; the resultant numbers are still a
> Pythagorean triple, and are relatively prime). Note that if any two of the
> triple have a common factor, the third must also have that as a factor.
>
> The PPTs [since at least one of a and b has to be even, choose it to be b]
> can be parametrized by the following:
>
> a = m^2 - n^2,
>
> b = 2mn,
>
> c = m^2 + n^2
>
> where m and n are relatively prime integers of opposite parity (if m is
> even, n is odd, and vice-versa). That is, every PPT is of this form, and
> every a,b,c of this form is a PPT. Since there are an infinite number of
> choices for m and n, there are an infinite number of PPTs. So the infinity
> of Pythagorean triples is not due to the existence of multiples of PPTs,
> but to the infinity of PPTs.
>
> Note also that not every Pythagorean triple can be parametrized in the form
> above, since (9,12,15) is a Pythagorean triple (= 3 times the PPT (3,4,5)),
> but 15 cannot be written as m^2+n^2 for any m,n. Thus, only PPTs can be
> parametrized in this way.
>
>
>
>
>
> mark snyder
>
>
>
> This is an unmoderated distribution list discussing post-calculus teaching
> and learning of mathematics.---David.Epstein@warwick.ac.uk
>
> Get guidelines before posting: email majordomo@warwick.ac.uk saying
> get mathedu guidelines
> (Un)subscribe to mathedu(-digest)by email to majordomo@warwick.ac.uk saying:
> (un)subscribe mathedu(-digest) <type in your email address here>
>
A. Jorge Garcia Teacher/Professor Mathematics/CompSci BaldwinSHS/NassauCC
=========================================================================
The Calculus Page http://freenet.buffalo.edu/~aj317
WorkBook Orders mailto:sffbookclub@compuserve.com
All Other EMail mailto:aj317@freenet.buffalo.edu
*************************************************************************
****************************************************************************
* To post to the list: email mathedcc@archives.math.utk.edu *
* To unsubscribe, send mail to: majordomo@archives.math.utk.edu *
* In the mail message, enter ONLY the words: unsubscribe mathedcc *
* Words in the Subject: line are NOT processed! *
* Archives at http://archives.math.utk.edu/hypermail/mathedcc/ *
****************************************************************************