$\underline{\text { Zhukov } \Gamma}$.

Let $C(n)$ be the number of distinct prime divisors of the number $n$. (For example, $C(10)=2, C(11)=1, C(12)=2$.)

Is the number of such pairs of natural numbers $(a, b)$, where $a \neq b$ and $C(a+b)=C(a)+C(b)$, finite or infinite?
For example, all pairs of the form $\left(2^{n}, 2^{n+1}\right)$ work. Here $C(a)=C(b)=1, C(a+b)=C\left(3 \cdot 2^{n}\right)=2$.

## Answer

Infinite.