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September 18th, 2013, 03:40 PM  #1 
Senior Member Joined: Sep 2012 From: British Columbia, Canada Posts: 764 Thanks: 53  Number of primes in an interval
For how many nonnegative integers n?25 does there exist a set of 1000 consecutive positive integers containing exactly n primes?

September 18th, 2013, 04:11 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Number of primes in an interval
I imagine this is true for all 26. It's true for 0 by the prime number theorem; let me hunt around for some numerics. Edit: It's also true for 1 by the prime number theorem. Otherwise, all primes would have a gap of less than a thousand, which would give them density >= n/998 which is much bigger than n/log n asymptotically. 
September 18th, 2013, 04:44 PM  #3 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Number of primes in an interval
The first example for n = 0 is {1693182318746372, ..., 1693182318747371}. I don't know when the first occurrence is for n = 1, but one example is {1693182318746352, ..., 1693182318747351}. The first example has all terms greater than 3*10^11. I don't know when the first occurrence is for n = 2, but one example is {1693182318746310, ..., 1693182318747309}. The first example has all terms greater than 3*10^9. I don't know when the first occurrence is for n = 3, but one example is {1686994940955704, ..., 1686994940956703}. The first example has all terms greater than 4*10^8. ... The first example for n = 25 is {172154114, ..., 172155113}. 
September 18th, 2013, 04:59 PM  #4 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: Number of primes in an interval
This is a nice question , i like it. 

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