Question 93
Let $\alpha$ be a real number and $a_1, a_2, a_3, \dots$ a strictly increasing sequence of positive integers such that for every $n \in \mathbb{N}$, $a_n \le n^\alpha$. A prime number $q$ is called *golden* if there is a positive integer $m$ such that $q \mid a_m$. Suppose that $q_1 \lt q_2 \lt q_3 \lt \dots$ are all *golden* prime numbers.
a) Prove that if $\alpha = 1.5$, then $q_n \le 1390^n$.
b) Prove that if $\alpha = 2.4$, then $q_n \le 1390^{2n}$.
a) Prove that if $\alpha = 1.5$, then $q_n \le 1390^n$.
b) Prove that if $\alpha = 2.4$, then $q_n \le 1390^{2n}$.