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quasi
Posts:
9,896
Registered:
7/15/05


Re:  question about coprimality
Posted:
Sep 17, 2011 3:52 AM


On Fri, 16 Sep 2011 17:42:31 0700 (PDT), Deep <deepkdeb@yahoo.com> wrote:
>Let a, b, c be three coprime integers such that 2b 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)(cb) = 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



