# Prove that (product of primes between n and 2n) > 2^n

• Nov 17th 2010, 06:28 AM
Newtonian
Prove that (product of primes between n and 2n) > 2^n
I've read that a weak but explicit version of the Prime Number Theorem states that the product of primes between $n$ and $2n$ (inclusive) is at least $2^n$ for all $n \geq 1$. I don't think this proof can be elementary (although if someone can prove it below then that would be great!) but is this a standard result? If so, where can I find a proof? I've searched but haven't managed to find anything.
• Nov 17th 2010, 03:23 PM
tonio
Quote:

Originally Posted by Newtonian
I've read that a weak but explicit version of the Prime Number Theorem states that the product of primes between $n$ and $2n$ (inclusive) is at least $2^n$ for all $n \geq 1$. I don't think this proof can be elementary (although if someone can prove it below then that would be great!) but is this a standard result? If so, where can I find a proof? I've searched but haven't managed to find anything.

Between 8 and 16 there are only two primes: 11 and 13, and $11\cdot 13 = 143 < 2^8 = 256$

Tonio
• Nov 17th 2010, 03:45 PM
Newtonian
That's a very good point...