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October 8th, 2010, 09:00 AM  #1 
Senior Member Joined: Apr 2008 Posts: 435 Thanks: 0  An interval with an odd number of primes
It has been a long while since I've started a topic, I think. Here is where I'm at: I've been looking for a way to quickly determine an interval containing an odd number of primes. That is, say I had the number K, and I wanted to find a number B such that [K,B] contained an odd number of primes. While it is true that I could simply start adding odd numbers, testing each with AKS, until I'm satisfied, I consider this too slow (actually, it's only probabilistic, and I am looking for something deterministic in nature). Do you have any thoughts? 
October 8th, 2010, 11:34 AM  #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: An interval with an odd number of primes
How long is your interval, and how large is the upper end? I have efficient solutions for the problem, depending on your answer. Also, are you finding just one or many? If many, then I can improve some solutions by precalculation. Also, see http://front.math.ucdavis.edu/1009.3956 for a recent exciting development relating to your problem (see section 2). 
October 17th, 2010, 02:37 AM  #3 
Senior Member Joined: Apr 2008 Posts: 435 Thanks: 0  Re: An interval with an odd number of primes
Actually, I am a contributor to polymath4 so I'm very familiar with that paper. This is one of the open problems that I thought I would ask. Sometimes, just one witty idea is all that is needed. Let's say we are given a number k. I seek to find an interval whose lower bound is k and whose upper bound is perhaps around 2k  that has an odd number of primes. Some thoughts might be to very wittily use arithmetic sequences that have high percentages of primes... or perhaps there are certain sets of intervals, say 5 different intervals, one of which will contain an odd number of primes. I am uncertain. Any ideas? 
October 17th, 2010, 01:09 PM  #4  
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: An interval with an odd number of primes Quote:
 

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