1

I am working in some calculations and I found this kind of problem. That is, are there infinitely many primes $p_n$ for which $p_n< 2n$ ? I think the answer is no, but I dont have a formal explanation.

GoldSoundz
  • 355
  • I don't know if I precisely understand what you mean, but there are only four primes (2,3,5,7) strictly smaller than $10 = 2 \cdot 5$. This seems to contradict your statement. – Demophilus Jun 05 '17 at 19:09
  • Demophilus , thank you for your answer. But could you prove that the statment holds only for the primes that you have indicated? – GoldSoundz Jun 05 '17 at 19:11

1 Answers1

5

First note that $p_5=11>2\cdot 5=10$.

And if $n\geq 5$ and $p_n>2n$, it follows that $$ p_{n+1}\geq p_n+2>2n+2=2(n+1) $$ so by induction $p_m>2m$ for all $m\geq 5$.

carmichael561
  • 53,688