20030906, 04:50  #1  
Sep 2003
3^{2}·7·41 Posts 
Incompatible HardyLittlewood conjectures: a DC project?
Most of us participating in GIMPS aren't mathematicians, but it's still fascinating to read about some of the mathematical background concerning primes, in Chris Caldwell's pages for instance.
Here's something probably wellknown to mathematicians, but new to me: the case of two "obvious" primenumber conjectures that cannot both be true: http://groups.google.com/groups?as_u...rbury.ac.nz%3E http://groups.google.com/groups?as_u...com%3E%231%2F1 (I found links to these Usenet postings at http://www.math.niu.edu/~rusin/known...dex/11NXX.html). This is also described more tersely at: http://mathworld.wolfram.com/HardyL...njectures.html The text of the second Usenet posting above (dated January 1999) states (with respect to finding a counterexample for the second "obvious" conjecture pi(x+y) <= pi(x) + pi(y)): Quote:
So I just wonder whether anyone ever thought of making a distributedcomputing project out of this. It would certainly be a much more significant result mathematically, if successful, than for instance the "Seventeen or Bust" project. Note I'm not personally proposing such a project, I'm just curious. By the way, one of the distributedcomputing projects listed in Aspenleaf involves calculation of pi(x) for large values of x, http://www.aspenleaf.com/distributed...projects.html http://www.aspenleaf.com/distributed...ixtableproject and it's fascinating to learn that there are algorithms for calculating pi(x) (the number of primes less than or equal to x) without actually obtaining a list of all such primes. Thus for instance we know that there are exactly 201,467,286,689,315,906,290 primes less than 10^22. http://numbers.computation.free.fr/C...ingPrimes.html Last fiddled with by ewmayer on 20031205 at 17:13 

20030914, 00:14  #2 
2·2,383 Posts 
Would finding a counterexample to the mentioned conjecture proove the other conjecture? If not the result won't be as significant. If it has been proved that one of the two is correct then the project would be much more useful.

20030914, 04:36  #3  
Sep 2003
3^{2}·7·41 Posts 
Quote:
However, the apparent consensus among most mathematicians is that the first (twinprimes) conjecture is true, which means the second one must therefore be false. 

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