On Fri, 16 Sep 2011 17:42:31 -0700 (PDT), Deep <firstname.lastname@example.org> 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: