MathLabs

Problem 1

Let Z\mathbb{Z} be the set of integers. Determine all functions f ⁣:Z→Zf\colon\mathbb{Z}\to\mathbb{Z} such that, for all integers aa and bb, f(2a)+2f(b)=f(f(a+b)).f(2a) + 2f(b) = f(f(a + b)).
Step 1 of 7: Name the assertion
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In plain words

Giving the equation a name lets us refer to specific substitutions concisely.

P(a,b): f(2a)+2f(b)=f(f(a+b))P(a,b):\ f(2a)+2f(b)=f(f(a+b))
Detailed analysis

Let P(a, b) denote the assertion f(2a) + 2f(b) = f(f(a+b)) for integers a and b. We first note that f identically zero and f(x) = 2x + c for a fixed integer c both satisfy the equation, and we will show these are the only solutions by extracting more and more structure from P.