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

1. ## 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 $\displaystyle n$ and $\displaystyle 2n$ (inclusive) is at least $\displaystyle 2^n$ for all $\displaystyle n \geq 1$. 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.

2. 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 $\displaystyle n$ and $\displaystyle 2n$ (inclusive) is at least $\displaystyle 2^n$ for all $\displaystyle n \geq 1$. 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 $\displaystyle 11\cdot 13 = 143 < 2^8 = 256$

Tonio

3. That's a very good point...

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