Question 153
Define a sequence $a_{0}, a_{1}, a_{2}, \ldots$ in the following way: $a_{0}=0$, and for $n \geq 0$,
$$
a_{n+1}=a_{n}+5^{a_{n}} .
$$
Let $k$ be any positive integer. Prove that the remainders when $a_{0}, a_{1}, \ldots, a_{2^{k-1}}$ are divided by $2^{k}$ are all different.
$$
a_{n+1}=a_{n}+5^{a_{n}} .
$$
Let $k$ be any positive integer. Prove that the remainders when $a_{0}, a_{1}, \ldots, a_{2^{k-1}}$ are divided by $2^{k}$ are all different.