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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.

Grade:11

1 Answers

Ashwin Muralidharan IIT Madras
290 Points
10 years ago

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).

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