FIND ALL f(x) such that (f(m)+n)(f(n)+m) is a perfet square for all m,n natural numbers including 0.

Hi Aritra,

Note that when f(n) = n + λ, where λ is a positive integer, the product becomes

(m+λ+n)(n+λ+m) = (n+m+λ)², a perfect square.

Now in case if you need a proof (which is indeed quite very challenging, much beyond the difficulty level of IITs), that this is the only function for which, the given expression would be a perfect square, mail me at ashwin.rsmd@gmail.com and i will send you back the proof.

Best Regards,

Ashwin (IIT Madras).