Question:For any a,b∈N+, if a+b is a square number, then f(a)+f(b) is also a square number. Find all such functions.

My try:It is clear that the function

f(x)=x satisfies the given conditions, since:

f(a)+f(b)=a+b.But is it the only function that fits our needs?

It’s one of my friends that gave me this problem, maybe this is a Mathematical olympiad problem. Thank you for you help.

**Answer**

It’s not a complete answer, but as mentioned in comments, this problem probably missed some restrictions, and so have too many solutions. Thus I decided to answer this question for the case that f have constant value in infinite (or finite by little changes) partition of N.

I expect another answers for remained cases e.g when f is an increasing function (polynomial case mentioned in comments).

Let A is an infinite subset of N, not containing two numbers with square sum (like https://oeis.org/A203988 except elements of the form (2k)22 in this sequence) and A′=N−A . Suppose A1,A2,... is an infinite non-empty partition of A, now f could be defined as below

f(n)={a=(2k)22if n∈A′a21−aif n∈A1a22−aif n∈A2... where k and ai∈N .

Now if x,y∈N and x+y is a perfect square, then both of x and y should be contained in A′, or on of them is in A′ and another one is in A (and so contained in one of the Ai), in both cases f(x)+f(y) is a perfect square .

**Attribution***Source : Link , Question Author : math110 , Answer Author :
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