achille
Posts:
575
Registered:
2/10/09


Re: A possible square?
Posted:
Jun 19, 2011 1:16 PM


On Jun 20, 12:52 am, "alainvergh...@gmail.com" <alainvergh...@gmail.com> wrote: > Good afternoon, > > Can 35*a^29 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)(3bi). LHS contains a factor 7 which is prime over Z[i]. => 7  (3+bi)(3bi) over Z[i] => 7  (3+bi) OR 7 (3bi) 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.

