Problem

Source: Iran 3rd round 2011-final exam-p5

Tags: number theory, prime numbers



Suppose that $\alpha$ is a real number and $a_1<a_2<.....$ is a strictly increasing sequence of natural numbers such that for each natural number $n$ we have $a_n\le n^{\alpha}$. We call the prime number $q$ golden if there exists a natural number $m$ such that $q|a_m$. Suppose that $q_1<q_2<q_3<.....$ are all the golden prime numbers of the sequence $\{a_n\}$. a) Prove that if $\alpha=1.5$, then $q_n\le 1390^n$. Can you find a better bound for $q_n$? b) Prove that if $\alpha=2.4$, then $q_n\le 1390^{2n}$. Can you find a better bound for $q_n$? part a proposed by mahyar sefidgaran by an idea of this question that the $n$th prime number is less than $2^{2n-2}$ part b proposed by mostafa einollah zade