Question 155
Given the equation
$$
a^b \cdot b^c = c^a
$$
in positive integers $a$, $b$ and $c$.
(i) Prove that any prime divisor of $a$ divides $b$ as well.
(ii) Solve the equation under the assumption $b \ge a$.
(iii) Prove that the equation has infinitely many solutions.
$$
a^b \cdot b^c = c^a
$$
in positive integers $a$, $b$ and $c$.
(i) Prove that any prime divisor of $a$ divides $b$ as well.
(ii) Solve the equation under the assumption $b \ge a$.
(iii) Prove that the equation has infinitely many solutions.