Date: Sep 17, 2011 7:30 AM
Author: Deep Deb
Subject: Re: ------------- question about coprimality

On Sep 17, 4:36 am, quasi <qu...@null.set> wrote:
> On Fri, 16 Sep 2011 17:42:31 -0700 (PDT), Deep <deepk...@yahoo.com>
> wrote:
>

> >Let a, b, c be three coprime integers such that 2|b and none
> >is a prime. Further let a = de where d and e are coprime. Now
> >consider (1)

>
> >c^2 - b^2 = a^2               (1).  
>
> >From (1) one gets (2) and then (2.1) and (2.2)
>
> >(c + b )(c - b) = (de)^2     (2);    
>
> >c + b = d^2       (2.1)
> >c - b = e^2       (2.2)

>
> >Question: Are (2.1) and (2.2) unique ? If not why not ?
>
> No -- there are lots of counterexamples.
>
> For example, if a=33 the equation
>
>    (c+b)(c-b) = a^2
>
> has more than one solution satisfying your conditions:
>
>    c=65, b=56 (which corresponds to d=3, e=11)
>
>    c=545, b=544 (which corresponds to d=1, e=33)
>
> quasi


----- ----- ------
You are right. My question is if (c+b),(c-b) are coprime. Then it
follows
c+b = d and c - b = e where d, e are coprime and both are odd. d and e
need not be unique.
But (2.1) and (2.2) are valid but not unique.
You may like to comment.
---- ---- ---- -----