Question 16
Let $N \gt 2^{5000}$ be a positive integer. Prove that if $1 \le a_1 \lt \dots \lt a_k \lt 100$ are distinct positive integers then the number
$$
\prod_{i=1}^{k} (N^{a_i} + a_i)
$$
has at least $k$ distinct prime factors.
$$
\prod_{i=1}^{k} (N^{a_i} + a_i)
$$
has at least $k$ distinct prime factors.