October 20th, 2015, 11:46 AM  #1 
Banned Camp Joined: Dec 2013 Posts: 1,117 Thanks: 41  Twin primes conjecture
Hi, r=int(sqrt(n)) Conjecture : For each n>2 there is always at least one pair of twinprime numbers between n!+n (not included) and n!+(r+1)^n (included)? Any counterexample? Is there a way to prove it if the conjecture is true? 
October 20th, 2015, 11:58 AM  #2 
Member Joined: Jun 2015 From: Ohio Posts: 99 Thanks: 19 
It isn't even known if there are infinitely many twin primes, so you would probably need to prove that before you were able to tackle this. That has been unproved for quite a long time, so I assume the problem is very challenging.

October 20th, 2015, 01:14 PM  #3 
Banned Camp Joined: Dec 2013 Posts: 1,117 Thanks: 41 
I assume that there are infinite twin primes. Trying to find a thin range where there at least one prime we can consider that as n goes to infinite (r+1)^n is far less < n!+n. So m=(n!+(r+1)^n)/(n!+n) will be very < 2. m will surely converges to some value between 1 and 2. My goal as I said before if to find a thin range where there is at least one prime. I do not think that my conjecture will work for a twin prime but for a prime I will not discard it. Thank you for your comment. 
October 20th, 2015, 09:57 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  
October 20th, 2015, 10:00 PM  #5 
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  You can do that, and it's not unreasonable, but it doesn't help proving the conjecture at hand because (a priori) we don't know anything about how they might be distributed. Yes, it converges to 1. 
October 21st, 2015, 12:13 PM  #6 
Banned Camp Joined: Dec 2013 Posts: 1,117 Thanks: 41 
Hard to prove but not impossible.

October 21st, 2015, 01:13 PM  #7 
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  

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conjecture, primes, twin 
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