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

Question

Ashwin Muralidharan IIT Madras · Answer

Hi Aritra, Note that when f(n) = n + &lambda;, where &lambda; is a positive integer, the product becomes (m+&lambda;+n)(n+&lambda;+m) = (n+m+&lambda;) 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).

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

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

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