Problem 5

Determine all functions

f:\mathbb{Z}\to\mathbb{Z}

such that

f(f(a)-b)+b\,f(2a)

is a perfect square for all integers

a

and

b

Step 3 of 5: Verify the zero-on-evens family

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f(2a)=0\ \Rightarrow\ f(f(a)-b)+bf(2a)=f(f(a)-b),\ \text{a square for every value }f(a)-b

Detailed analysis

If $f$ vanishes on all even integers and equals a perfect square on each odd integer, then $2a$ is always even, so $f(2a)=0$ and the expression reduces to $f(f(a)-b)$ . As $b$ ranges over all integers, so does $m=f(a)-b$ , and by construction $f(m)$ is always a perfect square ( $0$ if $m$ is even, the chosen square if $m$ is odd). So the whole expression is a perfect square for every $a,b$ .