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Topic: A possible square?
Replies: 5   Last Post: Jun 20, 2011 4:28 AM

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achille

Posts: 575
Registered: 2/10/09
Re: A possible square?
Posted: Jun 19, 2011 1:16 PM
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On Jun 20, 12:52 am, "alainvergh...@gmail.com"
<alainvergh...@gmail.com> wrote:
> Good afternoon,
>
> Can 35*a^2-9 be a square?
> 'a'  a positive integer,
>
> Alain


No.

Assume the contrary. Let's say 35a^2 = 9 + b^2 for some
+ve integers a,b, Over Z[i], the gaussian integers, which
is known to be an unique factorization domain, we have:

RHS can be written as (3+bi)(3-bi).
LHS contains a factor 7 which is prime over Z[i].
=> 7 | (3+bi)(3-bi) over Z[i]
=> 7 | (3+bi) OR 7 |(3-bi) over Z[i]
=> 7 | 3 over Z, impossible!

With this as hint, one can construct a more elementary
proof by verifying the possible values of

9 + b^2 mod 7 can only be 2,3,4,6

while 35 a^2 mod 7 is always 0.



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