4. Find the smallest positive integer $k$, such that $\varphi(n)=k$ has no solution; exactly two solutions; exactly three solutions; exactly four solutions (an unsolved conjecture is: there does not exist a positive integer $k$, such that $\varphi(n)=$ $k$ has exactly one solution).


4. $k=3$ has no solution; $k=1$ has exactly two solutions: $\varphi(2)=\varphi(1)=1$, because $2 \mid \varphi(n), n \geqslant 3 ; k=2$ has exactly three solutions: $\varphi(6)=\varphi(4)=\varphi(3)=2$, because at this time $n$ can only have prime factors less than 5; $k=4$ has exactly four solutions: $\varphi(12)=\varphi(10)=\varphi(8)=\varphi(5)=4$, because at this time $n$ can only have prime factors less than 7 (using equation (5)).