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

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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.

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