Question 60
Let $a, b \gt 1$ be positive integers. Prove that there are infinitely many positive integers $n$ such that the following equation has no solution $(k, t)$ in positive integers.
$$
\varphi(a^n - 1) = b^k - b^t.
$$
$$
\varphi(a^n - 1) = b^k - b^t.
$$