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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: 234 Re: Number of primes in an interval This is a nice question , i like it.  Tags interval, number, primes Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post caters Number Theory 67 March 19th, 2014 04:32 PM icemanfan Number Theory 12 March 3rd, 2012 04:29 PM jason.spade Number Theory 3 October 17th, 2010 01:09 PM niravshah99 Number Theory 4 June 20th, 2010 02:53 AM honzik Number Theory 1 June 10th, 2009 02:58 PM

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