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Let $n_0$ be 1 or 3. If a multiplicative function $f$ satisfies $f(p+q-n_0) =f(p)+f(q)-f(n_0)$ for all primes $p$ and $q$, then $f$ is the identity function$f(n)=n$ or a constant function $f(n)=1$.

Multiplicative functions with $f(p+q-n_0) = f(p)+f(q)-f(n_0)$