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Old 2003-09-06, 04:50   #1
GP2
 
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Default Incompatible Hardy-Littlewood 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 well-known to mathematicians, but new to me: the case of two "obvious" prime-number 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/Hardy-L...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:
This problem is within computer reach. I don't
have the CPU resources anymore to continue with it.
I have no idea what algorithm he was using, but the author considered it computationally "within reach" with 1999-era computers and resources probably limited to a single computer lab within his company (RSA).

So I just wonder whether anyone ever thought of making a distributed-computing 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 distributed-computing 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 2003-12-05 at 17:13
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Old 2003-09-14, 00:14   #2
AntiMagicMan
 

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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.
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Old 2003-09-14, 04:36   #3
GP2
 
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Quote:
Originally posted by AntiMagicMan
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.
As far as I know, it's merely been proven that both cannot be true.

However, the apparent consensus among most mathematicians is that the first (twin-primes) conjecture is true, which means the second one must therefore be false.
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