MathLabs

Problem 5

Determine all functions f:Z→Zf:\mathbb{Z}\to\mathbb{Z} such that f(f(a)−b)+b f(2a)f(f(a)-b)+b\,f(2a) is a perfect square for all integers aa and bb.
Step 2 of 5: Verify f(n)=n^2
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f(f(a)−b)+bf(2a)=(a2+b)2 when f(n)=n2f(f(a)-b)+bf(2a)=(a^2+b)^2\ \text{when } f(n)=n^2
Detailed analysis

If f(n)=n2f(n)=n^2, then f(f(a)−b)+bf(2a)=(a2−b)2+b(2a)2=a4−2a2b+b2+4a2b=a4+2a2b+b2=(a2+b)2f(f(a)-b)+bf(2a)=(a^2-b)^2+b(2a)^2=a^4-2a^2b+b^2+4a^2b=a^4+2a^2b+b^2=(a^2+b)^2, a perfect square for every a,ba,b.