Math Question

Factual Questions
1

Question: Does there exist number, n, such that:
2…n has a lower percentage of prime numbers than does 2…2n?

The answer must be one of the following:

  1. There is a proof that there is no such number.
  2. There is a proof that there is such a number, but the number is not known.
  3. There is such a number and the number and it is_______________
  4. It is not known whether or not there is such a number.
2

See https://boards.straightdope.com/sdmb/showthread.php?t=850215 if you missed it the first time.

2½ is such a number.

3

There is proof that there is no such whole number.

As mentioned in the previous thread: For sufficiently large n there is proof that the density decreases, so the density primes in 2…n can’t be smaller than the same for 2…2n

And for smaller n we can just try all n’s:

In the free version I get that to show the ratio from 2-160, the closest it gets is 1.11 at n=36

And 160 is sufficiently large to be a sufficiently large n for the previously mentioned purpose.

4

Since the question has been completely answered, I will just throw in this ditty that the professor in a number theory course I took over 60 years ago:
Chebycheff proved it/ You can too/ There’s always a prime/ 'Tween x and x times 2". This is in fact true for x >1; x doesn’t have to be an integer.