if a^2 +b^2 +3ab is a square number.then what is (a,b)

Question

The whole function is a square number

Answer 1

a^2+b^2+3ab=a^2+b^2+2ab+ab=(a+b)^2+ab. If this is a perfect square then we can write (a+b)^2+ab=n^2.

Therefore ab=n^2-(a+b)^2 and n^2=(n-a-b)(n+a+b). This would be satisfied if a=n-a-b, and b=n+a+b.

So b=n-2a and a=-n, so b=n+2n=3n.

(a,b)=(-n,3n). a^2+b^2+3ab=n^2+9n^2-9n^2=n^2.

EXAMPLE: n=2: a=-2, b=6 and 4+36-36=4.