# Math Question

**URL:** <https://boards.straightdope.com/t/math-question/819620>\
**Category:** Factual Questions\
**Created:** [August 14, 2018, 9:51pm UTC](https://boards.straightdope.com/t/math-question/819620 "2018-08-14T21:51:58Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![Steven\_Estes](https://avatars.discourse-cdn.com/v4/letter/s/94ad74/32.png) [@Steven\_Estes](https://boards.straightdope.com/u/Steven_Estes)\
**Post date:** [August 14, 2018, 9:51pm UTC](https://boards.straightdope.com/t/math-question/819620/1 "2018-08-14T21:51:58Z")

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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.

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**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\
**Post date:** [August 14, 2018, 10:07pm UTC](https://boards.straightdope.com/t/math-question/819620/2 "2018-08-14T22:07:08Z")

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

2½ is such a number.

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**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\
**Post date:** [August 14, 2018, 11:15pm UTC](https://boards.straightdope.com/t/math-question/819620/3 "2018-08-14T23:15:01Z")

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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:

> **[pi(n)/pi(2n)\*(2n-1)/(n-1) - Wolfram|Alpha](https://www.wolframalpha.com/input/?i=pi%28n%29%2Fpi%282n%29*%282n-1%29%2F%28n-1%29)**
>
> Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

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.

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<div class="post-metadata">

**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\
**Post date:** [August 16, 2018, 12:10am UTC](https://boards.straightdope.com/t/math-question/819620/4 "2018-08-16T00:10:19Z")

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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.