primes as far as the eye can see, discrete continua
Saw in a recent _Science_ that Ben Green of Cambridge proved that for any N, there are an infinite number of evenly spaced progressions of primes that are N numbers long. He got a prize for that. Damn straight. Now back to the decline of the neo-roman empire...
On Tue, 7 Dec 2004, Major Variola (ret) wrote:
Saw in a recent _Science_ that Ben Green of Cambridge proved that for any N, there are an infinite number of evenly spaced progressions of primes that are N numbers long. He got a prize for that. Damn straight.
Where N is a natural number? How do they define "progression"? Depending on that definition, there are some trivial counter examples. -Chuck -- http://www.quantumlinux.com Quantum Linux Laboratories, LLC. ACCELERATING Business with Open Technology "The measure of the restoration lies in the extent to which we apply social values more noble than mere monetary profit." - FDR
What about where N=1? I don't understand. You can only have an infinite number (or number of progressions) where the number of numbers in a number is inifinite. -TD
From: "Major Variola (ret)" <mv@cdc.gov> To: "cypherpunks@al-qaeda.net" <cypherpunks@al-qaeda.net> Subject: primes as far as the eye can see, discrete continua Date: Tue, 07 Dec 2004 19:45:24 -0800
Saw in a recent _Science_ that Ben Green of Cambridge proved that for any N, there are an infinite number of evenly spaced progressions of primes that are N numbers long. He got a prize for that. Damn straight.
Now back to the decline of the neo-roman empire...
Tyler Durden wrote:
What about where N=1?
I don't understand. You can only have an infinite number (or number of progressions) where the number of numbers in a number is inifinite.
After googling up some references, it seems the Major made a small misstatement. Green appears to have proven that for any number N greater than 1, there are an infinite number of prime progressions where the primes are separated by N. For example, 3,5,7 are all primes differing by 2. The _Science_ article is behind their paid-subscription wall, so I can't look at the source, but http://www.sciencenews.org/articles/20040424/mathtrek.asp talks a bit about the general subject. -- Roy M. Silvernail is roy@rant-central.com, and you're not "It's just this little chromium switch, here." - TFT SpamAssassin->procmail->/dev/null->bliss http://www.rant-central.com
On 2004-12-08T10:30:22-0500, Tyler Durden wrote:
From: "Major Variola (ret)" <mv@cdc.gov>
Saw in a recent _Science_ that Ben Green of Cambridge proved that for any N, there are an infinite number of evenly spaced progressions of primes that are N numbers long. He got a prize for that.
What about where N=1?
I don't understand. You can only have an infinite number (or number of progressions) where the number of numbers in a number is inifinite.
True for N=1 trivially, because it's easily proven that there are infinitely many primes. (For a set of primes S, find the product of them all and add 1. The result is obviously not divisible by any prime in S, so it's either a prime or a composite that factors into at least two smaller primes not in S. Either way, add the new prime(s) to S, and repeat.) I looked at B. Green's paper, but got lost around page 10 (of 50). He apparently proves that there are arbitrarily long progressions of primes. From that, you can cut some such arbitrarily long progression of primes into k-length progressions, and as N->infinity, you end up approaching an infinite number of k-length progressions. It's even easier (conceptually) if you accept two different progressions that have different spacing. for instance, when N=3, 5,11,17 17,23,29 31,37,43 would be a set of equal-spacing progressions. 5,11,17 17,53,89 would be a set of unequal-spacing progressions. Different progressions have different spacings. The paper was giving me a headache so I don't want to try to figure out which he meant. Clearly, the former is stronger.
On 2004-12-08T11:10:28-0500, Roy M. Silvernail wrote:
Tyler Durden wrote:
What about where N=1?
I don't understand. You can only have an infinite number (or number of progressions) where the number of numbers in a number is inifinite.
differing by 2. The _Science_ article is behind their paid-subscription wall, so I can't look at the source, but
I'm not sure if this is the right paper, but it's what I was looking at: http://front.math.ucdavis.edu/math.NT/0404188 (linked from http://www.dpmms.cam.ac.uk/~bjg23/preprints.html)
participants (5)
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Chuck Wolber -
Justin -
Major Variola (ret) -
Roy M. Silvernail -
Tyler Durden