Note that 2007 is an odd number hence,
x and y must be of opposite parity, let x be of the form 2m+1 and y be of the form 2n. (Opposite parity).
hence the LHS becomes :
(2m+1)^2 + (2n)^2 =. 4m^2 + 4m +1 + 4n^2 = 4k+1 (for some integer k)
Therefore LHS is congruent to 1 mod 4 ...... But note that 2007 is congruent to 3 mod 4 ....contradiction !
Hence there are no integral solutions for this equation...Please correct me if my solution is wrong.
Thank you
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