20171228, 17:45  #67  
Jun 2003
Oxford, UK
1,901 Posts 
Quote:
Maybe this will be the next challenge for the PGS when we get to 2^64. 

20180101, 00:52  #68 
Dec 2017
2×5^{2} Posts 
I love how we all think the prime numbers are random when really....
I don’t think a lot of people know about the Prime Gap Equation but I found it on a Wikipedia Article and it just shows that every prime number determines the following prime number. So, WHAT ARE WE ALL TALKING ABOUT?! heh. And I read a book by Australian mathematician and standup comedian Matt Parker, called “Things to Make and Do in the Fourth Dimension” and he says that for some prime number p, there exists another prime number q that ranges from (p, p + 5414). Well, this is how I’m phrasing it, but he simply said in the book that each gap between two adjacent (neighbouring) prime numbers have an upper bound of 5414.
Last fiddled with by George M on 20180101 at 00:56 Reason: Because people need to know about the bound of prime gaps. 
20180101, 02:08  #69  
"Curtis"
Feb 2005
Riverside, CA
4,409 Posts 
Quote:
For your own education, find the next prime after this number: 293703234068022590158723766104419463425709075574811762098588798217895728858676728143227 Hint: the next prime is more than 5414 greater than this number, which disproves the hooey you cite. 

20180101, 03:52  #70  
Jun 2003
4,721 Posts 
Quote:
Basically, there was a result that there are infinitely many prime pairs p,q such that the gap qp is bounded by a small number. They successively improved the upper bound on that gap (which at one point stood at 5414  See http://michaelnielsen.org/polymath1/...ime_gap_bounds) 

20180101, 11:39  #71 
"Dana Jacobsen"
Feb 2011
Bangkok, TH
2×3×151 Posts 
I don't think we need to add any more on the recent topic. Based on George's other posts today, I think it was an early New Year's celebration that included random posts to lots of threads.
On topic, 10 of the top 14 merits were found in 2017 including the top 5. Before I moved resources over to the PGS exhaustive search, I'd put a fair amount into smaller P1s, leading to a lot more largemerit finds. Gapcoin may do even more in 2018 given the popularity of cryptocoins these day 
20180101, 13:05  #72  
Dec 2017
32_{16} Posts 
Quote:


20180101, 14:02  #73 
Dec 2017
62_{8} Posts 
Prime Gap Hystory
On 13 May 2013, an upper bound of prime gaps was proven to be 63,374,611 (rounding to 70 million). This was done by Yitang (Tom) Zhang.
Then Tim Trudgian brought it down to 59,874,594 with Scott Morrison bringing it further down to 59,470,640 around late May. At 31 May however, it was brought down to 42,342,946. Then a mathematician called Terence Tao who learnt algebra at aged 3, completed his maths degree at aged 16, got a maths PhD and won a Fields Medal in 2006, brought down the bound to 42,342,924. Terence Tao is known as the “hypergenius” at maths with an IQ of 220 (world’s highest). He and another Fields Medalist, Tim Gower, then started an open project as part of Polymath where mathematicians could join together and collaborate to bring this bound down. As of 20 July 2013, the upper bound was brought down to 5414. Doesn’t sound hooey to me, but if you say so... 
20180101, 14:11  #74  
Nov 2008
2·3^{3}·43 Posts 
Quote:
In fact arbitrarily large gaps exist: n!+m is divisible by m for m ≤ n, so there are n1 consecutive composite numbers from n!+2 to n!+n. This gives a gap of size at least n. 

20180102, 07:19  #75  
Dec 2017
2·5^{2} Posts 
Quote:


20180103, 19:35  #76 
Dec 2008
you know...around...
2^{3}·3·5^{2} Posts 
The new Gapcoin discovery is a marvel. It's reminiscent of the Nyman gap of 1132.
To top it off: if the lefthand bounding prime was composite, it would expand to a gap of merit=46.71  which would have been an even more mindblowing result. And all that without the benefits of a large primorial. I've attached a graph that shows that the numbers in the primeless interval that are coprime to about 200# (the order of magnitude of the primes themselves) is even a bit above the average. The graph itself shows that Gapcoin indeed uses "random" numbers, that is to say, without using primorials to take advantage of cancelling out a lot of small factors. (I've just come up with the term "coprime profile" for it  catchy/appropriate?)  For comparison, the second graph shows the same for a gap that utilizes a primorial. Last fiddled with by mart_r on 20180103 at 19:41 
20180103, 19:57  #77  
Aug 2006
1011100101101_{2} Posts 
Quote:


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