Find all functions satisfying $\frac{1}{f(a)}+ \frac{1}{f(b)} = \frac{1}{f(c)} $ whenever $\frac{1}{a}+ \frac{1}{b} = \frac{1}{c} $

Question

Find all functions $f: \mathbb{N} \rightarrow \mathbb{N} $ satisfying $\frac{1}{f(a)}+ \frac{1}{f(b)} = \frac{1}{f(c)} $ whenever $\frac{1}{a}+ \frac{1}{b} = \frac{1}{c} \ a,b,c \in \mathbb{N} $

After talking through it with a friend, they believe there is a simple solution only a couple of lines long, but I have yet to spot the method. Any ideas?

Answer 1

Hint: Try to show that $f(ab) = af(b)$ for all $a,b\in\mathbb{N}$. This can be done through induction on $a$: Suppose $f(kb) = kf(b)$ for all $b\in\mathbb{N}$ and $k = 1,\dots,a-1$, and note that $$\frac{1}{ab} + \frac{1}{(a-1)\times ab} = \frac{1}{(a-1)\times b}. $$ Can you proceed from here?

Answer 2

Hint: first fix $c=1$, if we have $\frac{1}{a}+\frac{1}{b}=1$ then $b+a=ab$ implies $f(b)+f(a)=\frac{f(a)f(b)}{f(1)}$, this forces the function $f: N \to N$ to behave in a predictable way (but not everywhere, only at certain pairs of points!) But now if we let $c$ vary?