October 16th, 2015, 06:45 AM  #11 
Banned Camp Joined: Dec 2013 Posts: 1,117 Thanks: 41 
Building a NON TRIVIAL and INFINITE sequence of composites numbers where no element is divided by any p < B (B is bound equal for example to 10.000 (I mean the first prime < 10.000)) is interesting. It will be a big discovery. Last edited by mobel; October 16th, 2015 at 07:12 AM. 
October 16th, 2015, 07:12 AM  #12 
Member Joined: Jun 2015 From: Ohio Posts: 99 Thanks: 19 
Here's a conjecture I thought of that definitely seems interesting. It's held for the few random values I've checked... Of course I have yet to check any large values (I'll do that later). If $\displaystyle 2^k  2^j  1$ is composite for all j from j = 1 to k  1 then $\displaystyle 2^k  1$ is a Mersenne Prime. It's not an if and only if because not all Mersenne primes have this property, but what if having this property guarantees it is a Mersenne prime? 
October 16th, 2015, 07:14 AM  #13  
Banned Camp Joined: Dec 2013 Posts: 1,117 Thanks: 41  Quote:
 
October 16th, 2015, 07:17 AM  #14 
Member Joined: Jun 2015 From: Ohio Posts: 99 Thanks: 19 
? k>5 isn't a counterexample... k = 6 works

October 16th, 2015, 07:30 AM  #15 
Banned Camp Joined: Dec 2013 Posts: 1,117 Thanks: 41  Sorry. I did a mistake. When k=3 2^31=7 72=5 74=3 Both are prime not composite 2^31 is a Mersenne prime so you have to bound k 2^51 is also a Mersenne prime Anyway you have to bound k Last edited by mobel; October 16th, 2015 at 07:33 AM. 
October 16th, 2015, 07:45 AM  #16 
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  Yes, that observation is obvious. But you misunderstood my point  it's thinness, not thickness, that made the HeathBrown proof interesting. (Sorry, I reread my post and it was confusing. HeathBrown's sequence is somewhat thin  less than x^(2/3) numbers up to x. You're asking about much thinner sequences, and so that would be far harder in general.)
Last edited by CRGreathouse; October 16th, 2015 at 07:53 AM. 
October 16th, 2015, 07:46 AM  #17 
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  I'd be happy to try  I have some ideas in mind  but I'll need a definition of "NON TRIVIAL" first.

October 16th, 2015, 07:51 AM  #18  
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  Quote:
 
October 16th, 2015, 07:58 AM  #19 
Banned Camp Joined: Dec 2013 Posts: 1,117 Thanks: 41  
October 16th, 2015, 08:22 AM  #20 
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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2k1, conjecture, form, primes 
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