Question 50
Let $p$ be a prime. We say that a sequence of integers $\{z_n\}_{n=0}^{\infty}$ is a $p$-pod if for each $e \ge 0$, there is an $N \ge 0$ such that whenever $m \ge N$, $p^e$ divides the sum
$$
\sum_{k=0}^{m} (-1)^k \binom{m}{k} z_k.
$$
Prove that if both sequences $\{x_n\}_{n=0}^{\infty}$ and $\{y_n\}_{n=0}^{\infty}$ are $p$-pods, then the sequence $\{x_n y_n\}_{n=0}^{\infty}$ is a $p$-pod.
$$
\sum_{k=0}^{m} (-1)^k \binom{m}{k} z_k.
$$
Prove that if both sequences $\{x_n\}_{n=0}^{\infty}$ and $\{y_n\}_{n=0}^{\infty}$ are $p$-pods, then the sequence $\{x_n y_n\}_{n=0}^{\infty}$ is a $p$-pod.