Primes and Powers of 2

[darktreb]

I remember reading somewhere something about the number of primes up to a certain n being bounded the powers of 2. It was something like there are <= n primes between 2^(n-1) and 2^n (but of course that isn't true). Can someone remind me what the theorem actually was?

[i/3]

There are so many results about prime numbers.


Some theorems related to powers of two :

. p(n) <= 2^(2^(n-1)) (p(n) is the nth prime number)

. (prod (p <= n, p prime) p) <= 4^n

. Bertrand's theorem :

for each n there is a prime number between n and 2*n

[ComplexZeta]

That's true, but I don't think the first one is very useful, since the bound is ridiculously large. However, Erdos did use the second one in his proof of Bertrand's Postulate.