Find all functions ff such that if a+ba+b is a square, then f(a)+f(b)f(a)+f(b) is a square

Question:

For any a,bN+, 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=NA . Suppose A1,A2,... is an infinite non-empty partition of A, now f could be defined as below
f(n)={a=(2k)22if nAa21aif nA1a22aif nA2... where k and aiN .

Now if x,yN 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 :
2 revsuser217174

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