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Topic: ------------- question about coprimality
Replies: 2   Last Post: Sep 17, 2011 7:30 AM

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quasi

Posts: 10,208
Registered: 7/15/05
Re: ------------- question about coprimality
Posted: Sep 17, 2011 3:52 AM
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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 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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