Problem

Source: 2020 Taiwan TST Round 1 Mock Exam P3

Tags: number theory, Taiwan



Let $N>2^{5000}$ be a positive integer. Prove that if $1\leq a_1<\cdots<a_k<100$ are distinct positive integers then the number \[\prod_{i=1}^{k}\left(N^{a_i}+a_i\right)\]has at least $k$ distinct prime factors. Note. Results with $2^{5000}$ replaced by some other constant $N_0$ will be awarded points depending on the value of $N_0$. Proposed by Evan Chen