infinitely many Mersenne primes??? - Page 2

TauCeti · 2003-11-11, 11:16

But the "probability" reasonings all assume a 'random' distribution of the primes.

One quote here from R.C. Vaughan (February 1990):
"It is evident that the primes are randomly distributed but, unfortunately, we don't know what 'random' means."

Take for example the 2nd Hardy-Littlewood conjecture that is widely assumed to be FALSE. It states, that pi(x+y) <= pi(x) + pi(y) for all x,y >2 (with pi the prime counting function).

If that conjecture is FALSE, there are 'regions' where the primes are somewhat 'dense'. It is assumed, that there are infinitly many of those dense regions. IF that is the case AND the fermat/mersenne primes are not randomly scattered about dense und sparse regions, the "probability" argument for an infinitude of mersenne primes and the unlikeliness of another fermat prime may not be that strong anymore.

Some interestiong insights also at http://www.srcf.ucam.org/~dmo25/Primes/

only_human · 2003-11-15, 00:42

Philmore's explanation nicely clears up my confusion.

I wanted to append a link to Chris Caldwell's Prime Pages definition of a heuristic argument because it so nicely fits with what is contained in this thread.

nucleon · 2003-11-25, 10:07

In New Scientist this week (22nd Nov 2003) there was an interview of a mathematician by the name of Ben Longstaff. One of his goals was to prove there is an infinite number of Mersenne Primes.

It was a shame GIMPs wasn't mentioned as well.

-- Craig

kwstone · 2003-11-26, 04:27

Here's the URL of the article http://www.newscientist.com/opinion/...MKE?id=ns24221
Actually, Ben Longstaff is the journalist. The mathematician being interviewed is Dr. Marcus du Sautoy. And it's interesting that he describes his attempts to prove the infinitude of Mersenne primes as my "crazy project"!

Here is du Sautoy's own site http://www.maths.ox.ac.uk/~dusautoy/

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2003-11-11, 11:16	#12
TauCeti Mar 2003 Braunschweig, Germany 2·113 Posts	But the "probability" reasonings all assume a 'random' distribution of the primes. One quote here from R.C. Vaughan (February 1990): "It is evident that the primes are randomly distributed but, unfortunately, we don't know what 'random' means." Take for example the 2nd Hardy-Littlewood conjecture that is widely assumed to be FALSE. It states, that pi(x+y) <= pi(x) + pi(y) for all x,y >2 (with pi the prime counting function). If that conjecture is FALSE, there are 'regions' where the primes are somewhat 'dense'. It is assumed, that there are infinitly many of those dense regions. IF that is the case AND the fermat/mersenne primes are not randomly scattered about dense und sparse regions, the "probability" argument for an infinitude of mersenne primes and the unlikeliness of another fermat prime may not be that strong anymore. Some interestiong insights also at http://www.srcf.ucam.org/~dmo25/Primes/

2003-11-15, 00:42	#13
only_human "Gang aft agley" Sep 2002 7252₈ Posts	Philmore's explanation nicely clears up my confusion. I wanted to append a link to Chris Caldwell's Prime Pages definition of a heuristic argument because it so nicely fits with what is contained in this thread.

2003-11-25, 10:07	#14
nucleon Mar 2003 Melbourne 203₁₆ Posts	In New Scientist this week (22nd Nov 2003) there was an interview of a mathematician by the name of Ben Longstaff. One of his goals was to prove there is an infinite number of Mersenne Primes. It was a shame GIMPs wasn't mentioned as well. -- Craig

2003-11-26, 04:27	#15
kwstone Jun 2003 Shanghai, China 109 Posts	Here's the URL of the article http://www.newscientist.com/opinion/...MKE?id=ns24221 Actually, Ben Longstaff is the journalist. The mathematician being interviewed is Dr. Marcus du Sautoy. And it's interesting that he describes his attempts to prove the infinitude of Mersenne primes as my "crazy project"! Here is du Sautoy's own site http://www.maths.ox.ac.uk/~dusautoy/