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?
Enter Fermat (FLT) stating that there are now solutions to a^n+b^n=c^n
where a,b,c are unequal positive integers and n is a positive integer
greater than 2.
So, if the exponents are all equal to 2, there are infiniely many
solutions (mutually prime or not) to these diophantine equtions. Further,
if the exponents of the diophantine equation are all equal but greater
than 2, no solution triplets exist.
Now Beal states: if a^x+b^y=c^z has a solution, then a,b and c have a
non-trivial common factor (GCF>1) where x,y and z are unequal integer
exponents greater than 2 and a,b,c are positive integers. In other words,
when the exponents are greater than 2 and unequal (FLT) all NO solution
triplets are mutually prime. I read somewhere that these triplets are
proven to be finite.
I wrote a program to try to find a counter example to Beal but was
unsuccessful. All the triplets I found had a GCF of 2 or 3!
Regards,
> On Thu, 27 Nov 1997, RWW Taylor wrote:
>
> >
> > > Given A,B,C,x,y,z whole numbers, x,y,z > 2. If A^x + B^y = C^z, then A, B, C
> >
> > > have a common factor.
> > >
> > > The only examples I could generate in a few minutes of playing with a
> > > spreadsheet are 32^3 + 8^5 = 16^4, and 15^4 + 15^5 = 30^4.
> > >
> >
> > There's a pattern there that could be exploited to generate an infinite number
> > of examples of a certain type -- for instance 31^5 + 31^6 = 62^6 and (working
> > backwards) the simpler example 7^3 + 7^4 = 14^3 . But that hardly takes us in
> > the direction of a solution. On the whole this problem just does not seem as
> > _compelling_ as FLT. But it is still of interest to note that
> > easily-understood conjectures that are surprisingly deep still lurk around the
> > corner -- everything has not yet been taken beyond everyday reach.
> >
> > RWW Taylor
> > National Technical Institute for the Deaf
> > Rochester Institute of Technology
> > Rochester NY 14623
> >
> > >>>> The plural of mongoose begins with p. <<<<
> >
> >
> > ****************************************************************************
> > * 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/ *
> > ****************************************************************************
> >
>
> 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
> *************************************************************************
>
>
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/ *
****************************************************************************