5
$\begingroup$

It seems to be the case that for any integer $x > 1 $, we have $ x \leq 2^{\pi(x)} $

I'm not sure whether this is obviously true, annoyingly false or difficult to prove. I'm hoping it's the former, and that I'm just being dim. I'm thinking along the lines of prime factorisation of $x$, but I can't see it. Any help would be greatly appreciated.

Thanks

  • 0
    Can you use the prime number theorem?2011-10-21
  • 0
    No. This has arisen as part of a question I'm trying to do. If it's not something elementary, then I must be approaching my original problem the wrong way.2011-10-21
  • 0
    From Bertrand's postulate we have $\pi(2^n) \ge n$ (by induction on $n$).2011-10-21
  • 0
    You don't need the full prime number theorem; [Bertrand's postulate](http://en.wikipedia.org/wiki/Bertrand%27s_postulate) will suffice. Of course that is not quite elementary either.2011-10-21

2 Answers 2