Let $f$ be an analytic function on the disc $D = \{z \in \mathbb{C} \; |\; |z| < 1\}$ satisfying $f(0) = 1$. Is the following statement true or false? If $f'(a) = f(a)$ whenever $\frac{1+a}{a}$ and $\frac{1−a}{a}$ are prime numbers, then $f(z) = e^z \; \forall z \in D$.
I have tried to write out the power series of both $f$ and $f'$ and matching them term by term. What is the point of the two prime numbers in this question? Thanks.