
Number Theory Number Theory Math Forum 
 LinkBack  Thread Tools  Display Modes 
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: 234  Re: Number of primes in an interval
This is a nice question , i like it. 

Tags 
interval, number, primes 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
primes and twin primes: Number between powers of 10  caters  Number Theory  67  March 19th, 2014 04:32 PM 
number of primes in a given interval  icemanfan  Number Theory  12  March 3rd, 2012 04:29 PM 
An interval with an odd number of primes  jason.spade  Number Theory  3  October 17th, 2010 01:09 PM 
Why primes number when doing modulo?  niravshah99  Number Theory  4  June 20th, 2010 02:53 AM 
Irrational number loops through the whole interval  honzik  Number Theory  1  June 10th, 2009 02:58 PM 